Algebraic data types: Difference between revisions

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Latest revision as of 05:15, 8 November 2023

Some languages offer direct support for algebraic data types and pattern matching on them. While this of course can always be simulated with manual tagging and conditionals, it allows for terse code which is easy to read, and can represent the algorithm directly.

Task

As an example, implement insertion in a red-black-tree.

A red-black-tree is a binary tree where each internal node has a color attribute red or black. Moreover, no red node can have a red child, and every path from the root to an empty node must contain the same number of black nodes. As a consequence, the tree is balanced, and must be re-balanced after an insertion.

Reference

Red-Black Trees in a Functional Setting

Bracmat

( ( balance
  =   a x b y c zd
    .       !arg
          : ( B
            .   ( ( R
                  .   ((R.?a,?x,?b),?y,?c)
                    | (?a,?x,(R.?b,?y,?c))
                  )
                , ?zd
                )
              | ( ?a
                , ?x
                , ( R
                  .   ((R.?b,?y,?c),?zd)
                    | (?b,?y,(R.?c,?zd))
                  )
                )
            )
        & (R.(B.!a,!x,!b),!y,(B.!c,!zd))
      | !arg
  )
& ( ins
  =   C X tree a m z
    .     !arg:(?X.?tree)
        & !tree:(?C.?a,?m,?z)
        & (   !X:<!m
            & balance$(!C.ins$(!X.!a),!m,!z)
          |   !X:>!m
            & balance$(!C.!a,!m,ins$(!X.!z))
          | !tree
          )
      | (R.,!X,)
  )
& ( insert
  =   X tree
    .   !arg:(?X.?tree)
      & ins$(!X.!tree):(?.?X)
      & (B.!X)
  )
& ( insertMany
  =   L R tree
    .     !arg:(%?L_%?R.?tree)
        & insertMany$(!L.!tree):?tree
        & insertMany$(!R.!tree)
      | insert$!arg
  )
);

Test:

(   (   it allows for terse code which is easy to read
      , and can represent the algorithm directly
    .
    )
  : ?values
& insertMany$(!values.):?tree
& lst$tree
& done
);

Output:

(tree=
  B
.   ( B
    .   (R.(B.,,),algorithm,(B.,allows,))
      , and
      , (B.,can,)
    )
  , code
  , ( R
    .   ( B
        .   (B.(R.,directly,),easy,)
          , for
          , (B.(R.,is,),it,)
        )
      , read
      , ( B
        .   (B.,represent,)
          , terse
          , (R.(B.,the,),to,(B.,which,))
        )
    )
);

C++

Translation of: Haskell

Compile time

C++ templates have a robust pattern matching facility, with some warts - for example, nested templates cannot be fully specialized, so we must use a dummy template parameter. This implementation uses C++17 deduced template parameters for genericity.

enum Color { R, B };
template<Color, class, auto, class> struct T;
struct E;

template<Color col, class a, auto x, class b> struct balance {
    using type = T<col, a, x, b>;
};
template<class a, auto x, class b, auto y, class c, auto z, class d>
struct balance<B, T<R, T<R, a, x, b>, y, c>, z, d> {
    using type = T<R, T<B, a, x, b>, y, T<B, c, z, d>>;
};
template<class a, auto x, class b, auto y, class c, auto z, class d>
struct balance<B, T<R, a, x, T<R, b, y, c>>, z, d> {
    using type = T<R, T<B, a, x, b>, y, T<B, c, z, d>>;
};
template<class a, auto x, class b, auto y, class c, auto z, class d>
struct balance<B, a, x, T<R, T<R, b, y, c>, z, d>> {
    using type = T<R, T<B, a, x, b>, y, T<B, c, z, d>>;
};
template<class a, auto x, class b, auto y, class c, auto z, class d>
struct balance<B, a, x, T<R, b, y, T<R, c, z, d>>> {
    using type = T<R, T<B, a, x, b>, y, T<B, c, z, d>>;
};

template<auto x, class s> struct insert {
    template<class, class = void> struct ins;
    template<class _> struct ins<E, _> { using type = T<R, E, x, E>; };
    template<Color col, class a, auto y, class b> struct ins<T<col, a, y, b>> {
        template<int, class = void> struct cond;
        template<class _> struct cond<-1, _> : balance<col, typename ins<a>::type, y, b> {};
        template<class _> struct cond<1, _> : balance<col, a, y, typename ins<b>::type> {};
        template<class _> struct cond<0, _> { using type = T<col, a, y, b>; };
        using type = typename cond<x < y ? -1 : y < x ? 1 : 0>::type;
    };
    template<class> struct repaint;
    template<Color col, class a, auto y, class b>
    struct repaint<T<col, a, y, b>> { using type = T<B, a, y, b>; };
    using type = typename repaint<typename ins<s>::type>::type;
};
template<auto x, class s> using insert_t = typename insert<x, s>::type;

template<class> void print();
int main() {
    print<insert_t<1, insert_t<2, insert_t<0, insert_t<4, E>>>>>();
}

Run time

Although C++ has structured bindings and pattern matching through function overloading, it is not yet possible to use them together so we must match the structure of the tree being rebalanced separately from decomposing it into its elements. A further issue is that function overloads are not ordered, so to avoid ambiguity we must explicitly reject any (ill-formed) trees that would match more than one case during rebalance.

#include <memory>
#include <variant>

template<class... Ts> struct overloaded : Ts... { using Ts::operator()...; };
template<class... Ts> overloaded(Ts...) -> overloaded<Ts...>;

enum Color { R, B };
using E = std::monostate;
template<class, Color> struct Node;
template<class T, Color C> using Ptr = std::unique_ptr<Node<T, C>>;
template<class T> using Tree = std::variant<E, Ptr<T, R>, Ptr<T, B>>;
template<class T, Color Col> struct Node {
    static constexpr auto C = Col;
    Tree<T> left;
    T value;
    Tree<T> right;
};
template<Color C, class A, class T, class B> Tree<T> tree(A&& a, T& x, B&& b) {
    return Tree<T>{Ptr<T, C>{new Node<T, C>{std::move(a), std::move(x), std::move(b)}}};
}

template<class T> Tree<T> balance(Tree<T> s) {
    auto&& ll = [](Ptr<T, R>& s, Ptr<T, R>& t, auto&, Ptr<T, B>& u, auto&, auto&, auto&) {
        auto& [a, x, b] = *s;
        auto& [s_, y, c] = *t;
        auto& [t_, z, d] = *u;
        return tree<R>(tree<B>(a, x, b), y, tree<B>(c, z, d));
    };
    auto&& lr = [](auto&, Ptr<T, R>& s, Ptr<T, R>& t, Ptr<T, B>& u, auto&, auto&, auto&) {
        auto& [a, x, t_] = *s;
        auto& [b, y, c] = *t;
        auto& [s_, z, d] = *u;
        return tree<R>(tree<B>(a, x, b), y, tree<B>(c, z, d));
    };
    auto&& rl = [](auto&, auto&, auto&, Ptr<T, B>& s, Ptr<T, R>& t, Ptr<T, R>& u, auto&) {
        auto& [a, x, u_] = *s;
        auto& [b, y, c] = *t;
        auto& [t_, z, d] = *u;
        return tree<R>(tree<B>(a, x, b), y, tree<B>(c, z, d));
    };
    auto&& rr = [](auto&, auto&, auto&, Ptr<T, B>& s, auto&, Ptr<T, R>& t, Ptr<T, R>& u) {
        auto& [a, x, t_] = *s;
        auto& [b, y, u_] = *t;
        auto& [c, z, d] = *u;
        return tree<R>(tree<B>(a, x, b), y, tree<B>(c, z, d));
    };
    auto&& l = [](auto& s) -> Tree<T>& {
        return *std::visit(overloaded{[&](E) { return &s; }, [](auto& t) { return &t->left; }}, s);
    };
    auto&& r = [](auto& s) -> Tree<T>& {
        return *std::visit(overloaded{[&](E) { return &s; }, [](auto& t) { return &t->right; }}, s);
    };
    return std::visit([&](auto&... ss) -> Tree<T> {
        if constexpr (1 <
            std::is_invocable_v<decltype(ll), decltype(ss)...> +
            std::is_invocable_v<decltype(lr), decltype(ss)...> +
            std::is_invocable_v<decltype(rl), decltype(ss)...> +
            std::is_invocable_v<decltype(rr), decltype(ss)...>)
            throw std::logic_error{""};
        else
            return overloaded{ll, lr, rl, rr, [&](auto&... ss) { return std::move(s); }}(ss...);
    }, l(l(s)), l(s), r(l(s)), s, l(r(s)), r(s), r(r(s)));
}
template<class T> Tree<T> ins(T& x, Tree<T>& s) {
    return std::visit(overloaded{
        [&](E) -> Tree<T> { return tree<R>(s, x, s); },
        [&](auto& t) {
            auto& [a, y, b] = *t;
            static constexpr auto Col = std::remove_reference_t<decltype(*t)>::C;
            return x < y ? balance(tree<Col>(ins(x, a), y, b)) :
                y < x ? balance(tree<Col>(a, y, ins(x, b))) :
                std::move(s);
        },
    }, s);
}
template<class T> Tree<T> insert(T x, Tree<T> s) {
    return std::visit(overloaded{
        [](E) -> Tree<T> { throw std::logic_error{""}; },
        [](auto&& t) {
            auto& [a, y, b] = *t;
            return tree<B>(a, y, b);
        }
    }, ins(x, s));
}

#include <iostream>
template<class T> void print(Tree<T> const& s, int i = 0) {
    std::visit(overloaded{
        [](E) {},
        [&](auto& t) {
            auto& [a, y, b] = *t;
            print(a, i + 1);
            std::cout << std::string(i, ' ') << "RB"[t->C] << " " << y << "\n";
            print(b, i + 1);
        }
    }, s);
}
int main(int argc, char* argv[]) {
    auto t = Tree<std::string>{};
    for (auto i = 1; i != argc; ++i)
        t = insert(std::string{argv[i]}, std::move(t));
    print(t);
}

C#

Translation of several

Works with: C sharp version 8

using System;

class Tree
{
    public static void Main() {
        Tree tree = Tree.E;
        for (int i = 1; i <= 16; i++) {
            tree = tree.Insert(i);
        }
        tree.Print();
    }

    private const bool B = false, R = true;
    public static readonly Tree E = new Tree(B, null, 0, null);

    private Tree(bool c, Tree? l, int v, Tree? r) => (IsRed, Left, Value, Right) = (c, l ?? this, v, r ?? this);

    public bool IsRed { get; private set; }
    public int Value { get; }
    public Tree Left { get; }
    public Tree Right { get; }

    public static implicit operator Tree((bool c, Tree l, int v, Tree r) t) => new Tree(t.c, t.l, t.v, t.r);
    public void Deconstruct(out bool c, out Tree l, out int v, out Tree r) => (c, l, v, r) = (IsRed, Left, Value, Right);
    public override string ToString() => this == E ? "[]" : $"[{(IsRed ? 'R' : 'B')}{Value}]";
    public Tree Insert(int x) => Ins(x).MakeBlack();
    private Tree MakeBlack() { IsRed = false; return this; }

    public void Print(int indent = 0) {
        if (this != E) Right.Print(indent + 1);
        Console.WriteLine(new string(' ', indent * 4) + ToString());
        if (this != E) Left.Print(indent + 1);
    }

    private Tree Ins(int x) => Math.Sign(x.CompareTo(Value)) switch {
         _ when this == E => (R, E, x, E),
        -1 => new Tree(IsRed, Left.Ins(x) , Value, Right).Balance(),
         1 => new Tree(IsRed, Left , Value, Right.Ins(x)).Balance(),
         _ => this
    };

    private Tree Balance() => this switch {
        (B, (R, (R, var a, var x, var b), var y, var c), var z, var d) => (R, (B, a, x, b), y, (B, c, z, d)),
        (B, (R, var a, var x, (R, var b, var y, var c)), var z, var d) => (R, (B, a, x, b), y, (B, c, z, d)),
        (B, var a, var x, (R, (R, var b, var y, var c), var z, var d)) => (R, (B, a, x, b), y, (B, c, z, d)),
        (B, var a, var x, (R, var b, var y, (R, var c, var z, var d))) => (R, (B, a, x, b), y, (B, c, z, d)),
        _ => this
    };
}

Output:

Clojure

Pattern matching library: core.match.

For code and a thorough write-up on the red-black tree implementation that uses core.match, please read: Clojure Cookbook - Data Structures: Red-Black Trees.

Common Lisp

Common Lisp doesn't come with any pattern-matching solutions on its own, but with the help of its macro facility, it can incorporate features from other languages such as pattern matching. Macros expand into efficient code during compilation time and there isn't much difference if it's included in the core language or not. As has been said, Lisp is a ball of mud and remains one no matter what one throws at it.

This is a straighforward translation of the TCL solution. I don't know red-black-trees myself but I tried mirroring the original program as closely as possible. It uses a pattern-matching library called toadstool.

Library: toadstool

(mapc #'use-package '(#:toadstool #:toadstool-system))
(defstruct (red-black-tree (:constructor tree (color left val right)))
  color left val right)
 
(defcomponent tree (operator macro-mixin))
(defexpand tree (color left val right)
  `(class red-black-tree red-black-tree-color ,color
                         red-black-tree-left ,left
                         red-black-tree-val ,val
                         red-black-tree-right ,right))
(pushnew 'tree *used-components*)
 
(defun balance (color left val right)
  (toad-ecase (color left val right)
    (('black (tree 'red (tree 'red a x b) y c) z d)
     (tree 'red (tree 'black a x b) y
           (tree 'black c z d)))
    (('black (tree 'red a x (tree 'red b y c)) z d)
     (tree 'red (tree 'black a x b) y (tree 'black c z d)))
    (('black a x (tree 'red (tree 'red b y c) z d))
     (tree 'red (tree 'black a x b) y (tree 'black c z d)))
    (('black a x (tree 'red b y (tree 'red c z d)))
     (tree 'red (tree 'black a x b) y (tree 'black c z d)))
    ((color a x b)
     (tree color a x b))))
 
(defun %insert (x s)
  (toad-ecase1 s
    (nil (tree 'red nil x nil))
    ((tree color a y b)
     (cond ((< x y)
            (balance color (%insert x a) y b))
           ((> x y)
            (balance color a y (%insert x b)))
           (t s)))))
 
(defun insert (x s)
  (toad-ecase1 (%insert x s)
    ((tree t a y b) (tree 'black a y b))))

E

Translation of: Haskell

In E, a pattern can be used almost anywhere a variable name can. Additionally, there are two operators used for pattern matching idioms: =~ (returns success as a boolean, somewhat like Perl's =~), and switch (matches multiple patterns, like Haskell's case).

Both of those operators are defined in terms of the basic bind/match operation: def pattern exit failure_handler := specimen

def balance(tree) {
  return if (
    tree =~ term`tree(black, tree(red, tree(red, @a, @x, @b), @y, @c), @z, @d)` ||
    tree =~ term`tree(black, tree(red, @a, @x, tree(red, @b, @y, @c)), @z, @d)` ||
    tree =~ term`tree(black, @a, @x, tree(red, tree(red, @b, @y, @c), @z, @d))` ||
    tree =~ term`tree(black, @a, @x, tree(red, @b, @y, tree(red, @c, @z, @d)))`
  ) {
    term`tree(red, tree(black, $a, $x, $b), $y, tree(black, $c, $z, $d))`
  } else { tree }
}
def insert(elem, tree) {
  def ins(tree) {
    return switch (tree) {
      match term`empty` { term`tree(red, empty, $elem, empty)` }
      match term`tree(@color, @a, @y, @b)` {
        if (elem < y) {
          balance(term`tree($color, ${ins(a)}, $y, $b)`)
        } else if (elem > y) {
          balance(term`tree($color, $a, $y, ${ins(b)})`)
        } else {
          tree
        }
      }
    }
  }
  def term`tree(@_, @a, @y, @b)` := ins(tree)
  return term`tree(black, $a, $y, $b)`
}

This code was tested by filling a tree with random values; you can try this at the E REPL:

? var tree := term`empty`
> for _ in 1..20 {
>   tree := insert(entropy.nextInt(100), tree)
> }
> tree

EchoLisp

;; code adapted from Racket and Common Lisp
;; Illustrates matching on structures
(require 'match)
(require 'struct)


(define (N-tostring n) (format "%s %d"  (N-color n) (N-value n)))
(struct N (color left value right) #:tostring N-tostring)

(define (balance t)
  (match t
    [(N '⚫️ (N '🔴 (N '🔴 a x b) y c) z d) (N '🔴 (N '⚫️ a x b) y (N '⚫️ c z d))]
    [(N '⚫️ (N '🔴 a x (N '🔴 b y c)) z d) (N '🔴 (N '⚫️ a x b) y (N '⚫️ c z d))]
    [(N '⚫️ a x (N '🔴 (N '🔴 b y c) z d)) (N '🔴 (N '⚫️ a x b) y (N '⚫️ c z d))]
    [(N '⚫️ a x (N '🔴 b y (N '🔴 c z d))) (N '🔴 (N '⚫️ a x b) y (N '⚫️ c z d))]
    [else t]))
    
    (define (ins value: x  tree: t)
    (match t
      ['empty (N '🔴 'empty x 'empty)]
      [(N c l v r) (cond [(< x v) (balance (N c (ins x l) v r))]
                         [(> x v) (balance (N c l v (ins x r)))]
                         [else t])]))
                         
	(define (insert value: x tree: s)
  		(match (ins x s) [(N _ l v r) (N '⚫️ l v r)]))

Output:

(define (t-show n (depth 0))
	(when (!eq? 'empty n)
		(t-show (N-left n) (+ 12 depth))
		(writeln (string-pad-left (format "%s" n ) depth))
		(t-show (N-right n) (+ 12 depth))))

(define T (for/fold [t 'empty] ([i 32]) (insert (random 100) t)))
(t-show T)

Elixir

Translation of: Erlang

But, it changed an API into the Elixir style.

defmodule RBtree do
  def find(nil, _), do: :not_found
  def find({ key, value, _, _, _ }, key), do: { :found, { key, value } }
  def find({ key1, _, _, left, _ }, key) when key < key1, do: find(left, key)
  def find({ key1, _, _, _, right }, key) when key > key1, do: find(right, key)
  
  def new(key, value), do: ins(nil, key, value) |> make_black
  
  def insert(tree, key, value), do: ins(tree, key, value) |> make_black
  
  defp ins(nil, key, value),
    do: { key, value, :r, nil, nil }
  defp ins({ key, _, color, left, right }, key, value),
    do: { key, value, color, left, right }
  defp ins({ ky, vy, cy, ly, ry }, key, value) when key < ky,
    do: balance({ ky, vy, cy, ins(ly, key, value), ry })
  defp ins({ ky, vy, cy, ly, ry }, key, value) when key > ky,
    do: balance({ ky, vy, cy, ly, ins(ry, key, value) })
  
  defp make_black({ key, value, _, left, right }),
    do: { key, value, :b, left, right }
  
  defp balance({ kx, vx, :b, lx, { ky, vy, :r, ly, { kz, vz, :r, lz, rz } } }),
    do: { ky, vy, :r, { kx, vx, :b, lx, ly }, { kz, vz, :b, lz, rz } }
  defp balance({ kx, vx, :b, lx, { ky, vy, :r, { kz, vz, :r, lz, rz }, ry } }),
    do: { kz, vz, :r, { kx, vx, :b, lx, lz }, { ky, vy, :b, rz, ry } }
  defp balance({ kx, vx, :b, { ky, vy, :r, { kz, vz, :r, lz, rz }, ry }, rx }),
    do: { ky, vy, :r, { kz, vz, :b, lz, rz }, { kx, vx, :b, ry, rx } }
  defp balance({ kx, vx, :b, { ky, vy, :r, ly, { kz, vz, :r, lz, rz } }, rx }),
    do: { kz, vz, :r, { ky, vy, :b, ly, lz }, { kx, vx, :b, rz, rx } }
  defp balance(t),
    do: t
end

RBtree.new(0,3)        |> IO.inspect
|> RBtree.insert(1,5)  |> IO.inspect
|> RBtree.insert(2,-1) |> IO.inspect
|> RBtree.insert(3,7)  |> IO.inspect
|> RBtree.insert(4,-3) |> IO.inspect
|> RBtree.insert(5,0)  |> IO.inspect
|> RBtree.insert(6,-1) |> IO.inspect
|> RBtree.insert(7,0)  |> IO.inspect
|> RBtree.find(4)      |> IO.inspect

Output:

{0, 3, :b, nil, nil}
{0, 3, :b, nil, {1, 5, :r, nil, nil}}
{1, 5, :b, {0, 3, :b, nil, nil}, {2, -1, :b, nil, nil}}
{1, 5, :b, {0, 3, :b, nil, nil}, {2, -1, :b, nil, {3, 7, :r, nil, nil}}}
{1, 5, :b, {0, 3, :b, nil, nil},
 {3, 7, :r, {2, -1, :b, nil, nil}, {4, -3, :b, nil, nil}}}
{1, 5, :b, {0, 3, :b, nil, nil},
 {3, 7, :r, {2, -1, :b, nil, nil}, {4, -3, :b, nil, {5, 0, :r, nil, nil}}}}
{3, 7, :b, {1, 5, :b, {0, 3, :b, nil, nil}, {2, -1, :b, nil, nil}},
 {5, 0, :b, {4, -3, :b, nil, nil}, {6, -1, :b, nil, nil}}}
{3, 7, :b, {1, 5, :b, {0, 3, :b, nil, nil}, {2, -1, :b, nil, nil}},
 {5, 0, :b, {4, -3, :b, nil, nil}, {6, -1, :b, nil, {7, 0, :r, nil, nil}}}}
{:found, {4, -3}}

Emacs Lisp

The pcase macro was added in Emacs 24.1. It's auto-loaded, so there's no need to add (require 'pcase) to your code.

(defun rbt-balance (tree)
  (pcase tree
    (`(B (R (R ,a ,x ,b) ,y ,c) ,z ,d) `(R (B ,a ,x ,b) ,y (B ,c ,z ,d)))
    (`(B (R ,a ,x (R ,b ,y ,c)) ,z ,d) `(R (B ,a ,x ,b) ,y (B ,c ,z ,d)))
    (`(B ,a ,x (R (R ,b ,y ,c) ,z ,d)) `(R (B ,a ,x ,b) ,y (B ,c ,z ,d)))
    (`(B ,a ,x (R ,b ,y (R ,c ,z ,d))) `(R (B ,a ,x ,b) ,y (B ,c ,z ,d)))
    (_                                 tree)))

(defun rbt-insert- (x s)
  (pcase s
    (`nil              `(R nil ,x nil))
    (`(,color ,a ,y ,b) (cond ((< x y)
                               (rbt-balance `(,color ,(rbt-insert- x a) ,y ,b)))
                              ((> x y)
                               (rbt-balance `(,color ,a ,y ,(rbt-insert- x b))))
                              (t
                               s)))
    (_                  (error "Expected tree: %S" s))))

(defun rbt-insert (x s)
  (pcase (rbt-insert- x s)
    (`(,_ ,a ,y ,b) `(B ,a ,y ,b))
    (_              (error "Internal error: %S" s))))

(let ((s nil))
  (dotimes (i 16)
    (setq s (rbt-insert (1+ i) s)))
  (pp s))

Output:

(B
 (B
  (B
   (B nil 1 nil)
   2
   (B nil 3 nil))
  4
  (B
   (B nil 5 nil)
   6
   (B nil 7 nil)))
 8
 (B
  (B
   (B nil 9 nil)
   10
   (B nil 11 nil))
  12
  (B
   (B nil 13 nil)
   14
   (B nil 15
      (R nil 16 nil)))))

Erlang

The code used here is extracted from Mark Northcott's GitHubGist.

-module(rbtree).
-export([insert/3, find/2]).
 
% Node structure: { Key, Value, Color, Smaller, Bigger }
 
find(_, nil) ->
  not_found;
find(Key, { Key, Value, _, _, _ }) ->
  { found, { Key, Value } };
find(Key, { Key1, _, _, Left, _ }) when Key < Key1 ->
  find(Key, Left);
find(Key, { Key1, _, _, _, Right }) when Key > Key1 ->
  find(Key, Right).
 
insert(Key, Value, Tree) ->
  make_black(ins(Key, Value, Tree)).
  
ins(Key, Value, nil) ->
  { Key, Value, r, nil, nil };
ins(Key, Value, { Key, _, Color, Left, Right }) ->
  { Key, Value, Color, Left, Right };
ins(Key, Value, { Ky, Vy, Cy, Ly, Ry }) when Key < Ky ->
  balance({ Ky, Vy, Cy, ins(Key, Value, Ly), Ry });
ins(Key, Value, { Ky, Vy, Cy, Ly, Ry }) when Key > Ky ->
  balance({ Ky, Vy, Cy, Ly, ins(Key, Value, Ry) }).
  
make_black({ Key, Value, _, Left, Right }) ->
  { Key, Value, b, Left, Right }.
  
balance({ Kx, Vx, b, Lx, { Ky, Vy, r, Ly, { Kz, Vz, r, Lz, Rz } } }) ->
  { Ky, Vy, r, { Kx, Vx, b, Lx, Ly }, { Kz, Vz, b, Lz, Rz } };
balance({ Kx, Vx, b, Lx, { Ky, Vy, r, { Kz, Vz, r, Lz, Rz }, Ry } }) ->
  { Kz, Vz, r, { Kx, Vx, b, Lx, Lz }, { Ky, Vy, b, Rz, Ry } };
balance({ Kx, Vx, b, { Ky, Vy, r, { Kz, Vz, r, Lz, Rz }, Ry }, Rx }) ->
  { Ky, Vy, r, { Kz, Vz, b, Lz, Rz }, { Kx, Vx, b, Ry, Rx } };
balance({ Kx, Vx, b, { Ky, Vy, r, Ly, { Kz, Vz, r, Lz, Rz } }, Rx }) ->
  { Kz, Vz, r, { Ky, Vy, b, Ly, Lz }, { Kx, Vx, b, Rz, Rx } };
balance(T) ->
  T.

Output:

> rbtree:insert(0,3,nil).
{0,3,b,nil,nil}
> T1 = rbtree:insert(0,3,nil).
{0,3,b,nil,nil}
> T2 = rbtree:insert(1,5,T1). 
{0,3,b,nil,{1,5,r,nil,nil}}
> T3 = rbtree:insert(2,-1,T2).
{1,5,b,{0,3,b,nil,nil},{2,-1,b,nil,nil}}
> T4 = rbtree:insert(3,7,T3). 
{1,5,b,{0,3,b,nil,nil},{2,-1,b,nil,{3,7,r,nil,nil}}}
> T5 = rbtree:insert(4,-3,T4).
{1,5,b,
 {0,3,b,nil,nil},
 {3,7,r,{2,-1,b,nil,nil},{4,-3,b,nil,nil}}}
> T6 = rbtree:insert(5,0,T5). 
{1,5,b,
 {0,3,b,nil,nil},
 {3,7,r,{2,-1,b,nil,nil},{4,-3,b,nil,{5,0,r,nil,nil}}}}
> T7 = rbtree:insert(6,-1,T6).
{3,7,b,
 {1,5,b,{0,3,b,nil,nil},{2,-1,b,nil,nil}},
 {5,0,b,{4,-3,b,nil,nil},{6,-1,b,nil,nil}}}
> T8 = rbtree:insert(7,0,T7). 
{3,7,b,
 {1,5,b,{0,3,b,nil,nil},{2,-1,b,nil,nil}},
 {5,0,b,{4,-3,b,nil,nil},{6,-1,b,nil,{7,0,r,nil,nil}}}}
> rbtree:find(4,T8).
{found,{4,-3}}

F#

// Pattern Matching. Nigel Galloway: January 15th., 2021
type colour= |Red |Black
type rbT<'N>= |Empty |N of colour * rbT<'N> * rbT<'N> * 'N
let repair=function |Black,N(Red,N(Red,ll,lr,lv),rl,v),rr,rv
                    |Black,N(Red,ll,N(Red,lr,rl,v),lv),rr,rv
                    |Black,ll,N(Red,N(Red,lr,rl,v),rr,rv),lv
                    |Black,ll,N(Red,lr,N(Red,rl,rr,rv),v),lv->N(Red,N(Black,ll,lr,lv),N(Black,rl,rr,rv),v)
                    |i,g,e,l->N(i,g,e,l)
let insert item rbt = let rec insert=function
                        |Empty->N(Red,Empty,Empty,item)
                        |N(i,g,e,l) as node->if item>l then repair(i,g,insert e,l) elif item<l then repair(i,insert g,e,l) else node
                      match insert rbt with N(_,g,e,l)->N(Black,g,e,l) |_->Empty

Go

Translation of: Kotlin

Go doesn't have algebraic data types as such though they can simulated (to a limited extent) by interfaces.

However, pattern matching on interfaces (via the type switch statement and type assertions) is limited to matching the implementing type and so the balance() method is not very pleasant.

package main

import "fmt"

type Color string

const (
    R Color = "R"
    B       = "B"
)

type Tree interface {
    ins(x int) Tree
}

type E struct{}

func (_ E) ins(x int) Tree {
    return T{R, E{}, x, E{}}
}

func (_ E) String() string {
    return "E"
}

type T struct {
    cl Color
    le Tree
    aa int
    ri Tree
}

func (t T) balance() Tree {
    if t.cl != B {
        return t
    }
    le, leIsT := t.le.(T)
    ri, riIsT := t.ri.(T)
    var lele, leri, rile, riri T
    var leleIsT, leriIsT, rileIsT, ririIsT bool
    if leIsT {
        lele, leleIsT = le.le.(T)
    }
    if leIsT {
        leri, leriIsT = le.ri.(T)
    }
    if riIsT {
        rile, rileIsT = ri.le.(T)
    }
    if riIsT {
        riri, ririIsT = ri.ri.(T)
    }
    switch {
    case leIsT && leleIsT && le.cl == R && lele.cl == R:
        _, t2, z, d := t.destruct()
        _, t3, y, c := t2.(T).destruct()
        _, a, x, b := t3.(T).destruct()
        return T{R, T{B, a, x, b}, y, T{B, c, z, d}}
    case leIsT && leriIsT && le.cl == R && leri.cl == R:
        _, t2, z, d := t.destruct()
        _, a, x, t3 := t2.(T).destruct()
        _, b, y, c := t3.(T).destruct()
        return T{R, T{B, a, x, b}, y, T{B, c, z, d}}
    case riIsT && rileIsT && ri.cl == R && rile.cl == R:
        _, a, x, t2 := t.destruct()
        _, t3, z, d := t2.(T).destruct()
        _, b, y, c := t3.(T).destruct()
        return T{R, T{B, a, x, b}, y, T{B, c, z, d}}
    case riIsT && ririIsT && ri.cl == R && riri.cl == R:
        _, a, x, t2 := t.destruct()
        _, b, y, t3 := t2.(T).destruct()
        _, c, z, d := t3.(T).destruct()
        return T{R, T{B, a, x, b}, y, T{B, c, z, d}}
    default:
        return t
    }
}

func (t T) ins(x int) Tree {
    switch {
    case x < t.aa:
        return T{t.cl, t.le.ins(x), t.aa, t.ri}.balance()
    case x > t.aa:
        return T{t.cl, t.le, t.aa, t.ri.ins(x)}.balance()
    default:
        return t
    }
}

func (t T) destruct() (Color, Tree, int, Tree) {
    return t.cl, t.le, t.aa, t.ri
}

func (t T) String() string {
    return fmt.Sprintf("T(%s, %v, %d, %v)", t.cl, t.le, t.aa, t.ri)
}

func insert(tr Tree, x int) Tree {
    t := tr.ins(x)
    switch t.(type) {
    case T:
        tt := t.(T)
        _, a, y, b := tt.destruct()
        return T{B, a, y, b}
    case E:
        return E{}
    default:
        return nil
    }
}

func main() {
    var tr Tree = E{}
    for i := 1; i <= 16; i++ {
        tr = insert(tr, i)
    }
    fmt.Println(tr)
}

Output:

T(B, T(B, T(B, T(B, E, 1, E), 2, T(B, E, 3, E)), 4, T(B, T(B, E, 5, E), 6, T(B, E, 7, E))), 8, T(B, T(B, T(B, E, 9, E), 10, T(B, E, 11, E)), 12, T(B, T(B, E, 13, E), 14, T(B, E, 15, T(R, E, 16, E)))))

Haskell

data Color = R | B
data Tree a = E | T Color (Tree a) a (Tree a)

balance :: Color -> Tree a -> a -> Tree a -> Tree a
balance B (T R (T R a x b) y c          ) z d                               = T R (T B a x b) y (T B c z d)
balance B (T R a           x (T R b y c)) z d                               = T R (T B a x b) y (T B c z d)
balance B a                               x (T R (T R b y c) z d          ) = T R (T B a x b) y (T B c z d)
balance B a                               x (T R b           y (T R c z d)) = T R (T B a x b) y (T B c z d)
balance col a x b = T col a x b

insert :: Ord a => a -> Tree a -> Tree a
insert x s = T B a y b where
  ins E          =  T R E x E
  ins s@(T col a y b) 
    | x < y      =  balance col (ins a) y b
    | x > y      =  balance col a y (ins b)
    | otherwise  =  s
  T _ a y b = ins s

J

J incorporates a symbol data type which, in versions 6.02 and 7.01, J implements directly as a red-black tree. The s: entry in the J dictionary begins Symbols are a data type and are created by the verb s:. Symbols provide a mechanism for searching, sorting, and comparisons more efficient than alternative mechanisms such as boxed strings. Structural, selection, and relational verbs work on symbols. Arithmetic verbs do not work on symbols. However, for this task we want to see an implementation and we want the implementation to represent an algebraic data type approach.

The following code represents a best effort translation of the current Haskell implementation of this task:

insert=:{{
  'R';'';y;a:
:
  if. 0=#y do. insert x
  elseif. 0=L.y do. x insert insert y
  else.
    'C e K w'=. y 
    select. *x - K
      case. _1 do. balance C;(x insert e);K;<w
      case.  0 do. y
      case.  1 do. balance C;e;K;<x insert w
    end.
  end.
}}

NB. C: color, e: east, K: key, w: west
NB. two cascaded reds under a black become two black siblings under a red 
balance=: {{
  'C e K w'=. y
  if. #e do.
    'eC ee eK ew'=. e
    if. 'R'=eC do.
      if. #ee do.
        'eeC eee eeK eew'=. ee NB. ((eee eeK eew) eK ew) K w   =>  (eee eeK eew) eK (ew K w)
        if. 'R'=eeC do. 'R';('B';eee;eeK;<eew);eK;<'B';ew;K;<w return. end. end.
      if. #ew do.
        'ewC ewe ewK eww'=. ew NB. (ee ek (ewe ewK eww)) K w  =>  (ee ek ewe) ewK (eww K w)
        if. 'R'=ewC do. 'R';('B';ee;eK;<ewe);ewK;<'B';eww;K;<w return. end. end. end. end.
  if. #w do.
    'wC we wK ww'=. w
    if. 'R'=wC do.
      if. #we do.
        'weC wee weK wew'=. we NB. e K ((wee weK wew) wK ww)  =>  (e K wee) weK (wew wK ww)
        if. 'R'=weC do. 'R';('B';e;K;<wee);weK;<'B';wew;wK;<ww return. end. end.
      if. #ww do.
        'wwC wwe wwK www'=. ww NB. e K (we wK (wwe wwK www))  =>  (e K we) wK (wwe wwK www) 
        if. 'R'=wwC do. 'R';('B';e;K;<we);wK;<'B';wwe;wwK;<www return. end. end. end. end.
  y
}}

Example use:

   3 insert 2 insert 5
┌─┬───────┬─┬───────┐
│R│┌─┬┬─┬┐│3│┌─┬┬─┬┐│
│ ││B││2│││ ││B││5│││
│ │└─┴┴─┴┘│ │└─┴┴─┴┘│
└─┴───────┴─┴───────┘

Note that by convention we treat the root node as black. This approach always labels it with 'R' which we ignore. However, if we wish to validate these trees, we must account for the discrepancy.

NB. always treat root of tree as black
validate=: {{
  if. 0=#y do. 1 return. end.
  'C e K w'=. y
  check 'B';e;K;<w
}}

check=: {{
  if. 0=#y do. 1 return. end.
  'C e K w'=. y
  if. 'R'=C do.
    if. 'R'={.;{.e do. 0 return. end.
    if. 'R'={.;{.w do. 0 return. end.
  end.
  a=. check e
  b=. check w
  (*a)*(a=b)*b+'B'=C
}}

Here, validate returns the effective "black depth" of the tree (treating the root node as black and treating empty nodes as black), or 0 if the tree is not balanced properly.

For example:

   ?.~20
14 18 12 16 5 1 3 0 6 13 9 8 15 17 2 10 7 4 19 11
   insert/?.~20
┌─┬──────────────────────────────────────────────────────────────────────┬──┬────────────────────────────────────────────────────────────────────────┐
│R│┌─┬───────────────────────────────────┬─┬────────────────────────────┐│10│┌─┬────────────────────────────────────────────────┬──┬────────────────┐│
│ ││R│┌─┬──────────────┬─┬──────────────┐│5│┌─┬───────┬─┬──────────────┐││  ││B│┌─┬────────────────┬──┬────────────────────────┐│17│┌─┬────────┬──┬┐││
│ ││ ││B│┌─┬┬─┬───────┐│2│┌─┬───────┬─┬┐││ ││B│┌─┬┬─┬┐│7│┌─┬┬─┬───────┐│││  ││ ││R│┌─┬┬──┬────────┐│13│┌─┬────────┬──┬────────┐││  ││B│┌─┬┬──┬┐│19││││
│ ││ ││ ││B││0│┌─┬┬─┬┐││ ││B│┌─┬┬─┬┐│4││││ ││ ││B││6│││ ││B││8│┌─┬┬─┬┐││││  ││ ││ ││B││11│┌─┬┬──┬┐││  ││B│┌─┬┬──┬┐│15│┌─┬┬──┬┐│││  ││ ││R││18│││  ││││
│ ││ ││ ││ ││ ││R││1││││ ││ ││R││3│││ ││││ ││ │└─┴┴─┴┘│ ││ ││ ││R││9││││││  ││ ││ ││ ││  ││R││12││││  ││ ││R││14│││  ││R││16│││││  ││ │└─┴┴──┴┘│  ││││
│ ││ ││ ││ ││ │└─┴┴─┴┘││ ││ │└─┴┴─┴┘│ ││││ ││ │       │ ││ ││ │└─┴┴─┴┘││││  ││ ││ ││ ││  │└─┴┴──┴┘││  ││ │└─┴┴──┴┘│  │└─┴┴──┴┘│││  │└─┴────────┴──┴┘││
│ ││ ││ │└─┴┴─┴───────┘│ │└─┴───────┴─┴┘││ ││ │       │ │└─┴┴─┴───────┘│││  ││ ││ │└─┴┴──┴────────┘│  │└─┴────────┴──┴────────┘││  │                ││
│ ││ │└─┴──────────────┴─┴──────────────┘│ │└─┴───────┴─┴──────────────┘││  ││ │└─┴────────────────┴──┴────────────────────────┘│  │                ││
│ │└─┴───────────────────────────────────┴─┴────────────────────────────┘│  │└─┴────────────────────────────────────────────────┴──┴────────────────┘│
└─┴──────────────────────────────────────────────────────────────────────┴──┴────────────────────────────────────────────────────────────────────────┘
   validate insert/?.~20
4

Finally a caution: red black trees exhibit poor cache coherency. In many (perhaps most or all) cases an amortized hierarchical linear sort mechanism will perform better than a red black tree implementation. (And that characteristic is especially true of this particular implementation.)

Java

Translation of: Kotlin

Works with: OpenJDK version 21 (Preview)

Java 21 has added support for ADTs (in the form of sealed types), which are narrowable through a switch expression. Despite having no fully-fledged pattern matching, a combination of record deconstruction patterns and guarded patterns allows for something very similar through switch expressions:

public class Task {
    enum Color { R, B }
    sealed interface Tree<A extends Comparable<A>> permits E, T {
        default Tree<A> insert(A a) {
            return switch(ins(a)) {
                case T(_, var l, var v, var r) -> new T<>(Color.B, l, v, r);
                case E() -> new E<>();
            };
        }

        Tree<A> ins(A a);
    }

    record E<A extends Comparable<A>>() implements Tree<A> {
        @Override
        public Tree<A> ins(A a) {
            return new T<>(Color.R, new E<>(), a, new E<>());
        }

        @Override
        public String toString() { return "E"; }
    }

    record T<A extends Comparable<A>>(Color color, Tree<A> left,
                                      A value, Tree<A> right) implements Tree<A> {
        @Override
        public Tree<A> ins(A a) {
            return switch(Integer.valueOf(a.compareTo(value))) {
                case Integer i when i < 0 -> new T<>(color, left.ins(a), value, right).balance();
                case Integer i when i > 0 -> new T<>(color, left, value, right.ins(a)).balance();
                default -> this;
            };
        }

        private Tree<A> balance() {
            if (color == Color.R) return this;
            return switch (this) {
                // unnamed patterns (case T<A>(_, ...)) are a JDK21 Preview feature
                case T<A>(_, T<A>(_, T<A>(_, var a, var x, var b), var y, var c), var z, var d)
                        when left instanceof T<A> le && le.left instanceof T<A> le_le &&
                             le.color == Color.R && le_le.color == Color.R ->
                                new T<>(Color.R, new T<>(Color.B, a, x, b), y, new T<>(Color.B, c, z, d));
                case T<A>(_, T<A>(_, var a, var x, T<A>(_, var b, var y, var c)), var z, var d)
                        when left instanceof T<A> le && le.right instanceof T<A> le_ri &&
                             le.color == Color.R && le_ri.color == Color.R ->
                                new T<>(Color.R, new T<>(Color.B, a, x, b), y, new T<>(Color.B, c, z, d));
                case T<A>(_, var a, var x, T<A>(_, T<A>(_, var b, var y, var c), var z, var d))
                        when right instanceof T<A> ri && ri.left instanceof T<A> ri_le &&
                             ri.color == Color.R && ri_le.color == Color.R ->
                                new T<>(Color.R, new T<>(Color.B, a, x, b), y, new T<>(Color.B, c, z, d));
                case T<A>(_, var a, var x, T<A>(_, var b, var y, T<A>(_, var c, var z, var d)))
                        when right instanceof T<A> ri && ri.right instanceof T<A> ri_ri &&
                             ri.color == Color.R && ri_ri.color == Color.R ->
                                new T<>(Color.R, new T<>(Color.B, a, x, b), y, new T<>(Color.B, c, z, d));
                default -> this;
            };
        }

        @Override
        public String toString() {
            return STR."T[\{color}, \{left}, \{value}, \{right}]"; // String templates are a JDK 21 Preview feature
        }
    }

    public static void main(String[] args) {
        Tree<Integer> tree = new E<>();
        for (var i : IntStream.rangeClosed(1, 16).toArray()) {
            tree = tree.insert(i);
        }
        System.out.println(tree);
    }
}

Output:

T[B, T[B, T[B, T[B, E, 1, E], 2, T[B, E, 3, E]], 4, T[B, T[B, E, 5, E], 6, T[B, E, 7, E]]], 8, T[B, T[B, T[B, E, 9, E], 10, T[B, E, 11, E]], 12, T[B, T[B, E, 13, E], 14, T[B, E, 15, T[R, E, 16, E]]]]]

jq

Adapted from Tcl

Works with: jq

Works with gojq, the Go implementation of jq

jq does not have built-in support for pattern matching in the sense of the present task description, but the following `bindings` function takes advantage of the way in which singleton-key JSON objects can be used as variables for pattern-matching. In effect, jq expressions such as `{a}` can be used as variables in the pattern definitions, and after matching, the corresponding values can be referenced by jq expressions such as `.a`.

Notice also how various features of jq come together to simplify the implementation of the `balance` function.

bindings.jq

# bindings($x) attempts to match . and $x structurally on the
# assumption that . is free of JSON objects, and that any objects in
# $x will have distinct, singleton keys that are to be interpreted as
# variables.  These variables will match the corresponding entities in
# . if . and $x can be structurally matched.
#
# If . and $x cannot be matched, then null is returned;
# otherwise, if $x contains no objects, {} is returned;
# finally, if . and $x can be structurally matched, a composite object containing the bindings
# will be returned.
# Output: null (failure to match) or a single JSON object giving the bindings if any.
def bindings($x):
   if $x == . then {}  # by assumption, no bindings are necessary
   elif ($x|type) == "object"
   then ($x|keys) as $keys
   | if ($keys|length) == 1 then {($keys[0]): .} else "objects should be singletons"|error end
   elif type != ($x|type) then null
   elif type == "array"
   then if length != ($x|length) then null
        else . as $in
        | reduce range(0;length) as $i ({};
            if . == null then null
            else ($in[$i] | bindings($x[$i]) ) as $m
            | if $m == null then null else . + $m end
            end)
	end
   else null
   end ;

pattern-matching.jq

include "bindings" {search: "."};

def E: [];  # the empty node
# Each nonempty node is an array: [Color, Left, Value, Right]
# where Left and Right are nodes.

def B: "⚫";
def R: "🔴";

def b(x): bindings({} | x) // empty;

# Input: [$color, $left, $value, $right]
def balance:
  def node: [R, [B, .a, .x, .b], .y, [B, .c, .z, .d]];

  (   b([B, [R, [R,  {a}, {x}, {x}], {y}, {c}],  {z}, {d}])
   // b([B, [R, {a}, {x}, [R,  {b},  {y}, {c}]], {z}, {d}])
   // b([B, {a},{x}, [R,  [R,  {b},  {y}, {c}],  {z}, {d}]])
   // b([B, {a},{x}, [R,  {b}, {y},  [R,  {c},   {z}, {d}]]])
   | node) // . ;

# Input: a node 
def ins($x):
  if . == E then [R, E, $x, E]
  else . as [$col, $left, $y, $right]
  | if   $x < $y then [ $col, ($left|ins($x)), $y, $right]            | balance
    elif $x > $y then [ $col, $left,           $y, ($right|ins($x)) ] | balance
    else $left
    end
  end;

# insert(Value) into .
def insert($x):
  ins($x) as [$col, $left, $y, $right]
  | [ B, $left, $y, $right] ;

def pp: walk( if type == "array" then map(select(length>0)) else . end);

def task($n):
  reduce range(0; $n) as $i (E; insert($i));

task(16) | pp

Output:

For brevity and perhaps visual appeal, the output from jq has been trimmed as per the following invocation:

jq -n -f pattern-matching.jq | grep -v '[][]' | tr -d ',"'

Julia

Julia's multiple dispatch model is based on the types of a function's arguments, but does not look deeper into the function's array arguments for the types of their contents. Therefore we do multi-dispatch on the balance function but then use an if statement within the multiply dispatched functions to further match based on argument vector contents.

import Base.length

abstract type AbstractColoredNode end

struct RedNode <: AbstractColoredNode end; const R = RedNode()
struct BlackNode <: AbstractColoredNode end; const B = BlackNode()
struct Empty end; const E = Empty()
length(e::Empty) = 1

function balance(b::BlackNode, v::Vector, z, d)
    if v[1] == R
        if length(v[2]) == 4 && v[2][1] == R
            return [R, [B, v[2][2], v[2][3], v[2][4]], v[3], [B, v[4], z, d]]
        elseif length(v[4]) == 4 && v[4][1] == R
            return [R, [B, v[2], v[3], v[4][2]], v[4][3], [B, v[4][4], z, d]]
        end
    end
    [b, v, z, d]
end

function balance(b::BlackNode, a, x, v::Vector)
    if v[1] == R
        if length(v[2]) == 4 && v[2][1] == R
            return [R, [B, a, x, v[2][2]], v[2][3], [B, v[2][4], v[3], v[4]]]
        elseif length(v[4]) == 4 && v[4][1] == R
            return [R, [B, a, x, v[2]], v[3], [B, v[4][2], v[4][3], v[4][4]]]
        end
    end
    [b, a, x, v]
end

function balance(b::BlackNode, a::Vector, x, v::Vector)
    if v[1] == R
        if length(v[2]) == 4 && v[2][1] == R
            return [R, [B, a, x, v[2][2]], v[2][3], [B, v[2][4], v[3], v[4]]]
        elseif length(v[4]) == 4 && v[4][1] == R
            return [R, [B, a, x, v[2]], v[3], [B, v[4][2], v[4][3], v[4][4]]]
        end
    end
    [b, a, x, v]
end

balance(node, l, a, r) = [node, l, a, r]

function ins(v::Vector, x::Number)
    if length(v) == 4
        if x < v[3]
            return balance(v[1], ins(v[2], x), v[3], v[4])
        elseif x > v[3]
            return balance(v[1], v[2], v[3], ins(v[4], x))
        end
    end
    v
end

ins(t, a) = [R, E, a, E]

insert(v, a) = (t = ins(v, a); t[1] = B; t)

function testRB()
    t = E
    for i in rand(collect(1:20), 10)
        t = insert(t, i)
    end
    println(replace(string(t), r"lackNode\(\)|edNode\(\)|Any|mpty\(\)" => ""))
end

testRB()

Output:

[B, [R, [B, [R, E, 1, E], 2, [R, E, 3, E]], 4, [B, E, 6, E]], 14, [B, E, 18, E]]]

Kotlin

Translation of: Scala

Whilst Kotlin supports algebraic data types (via 'sealed classes') and destructuring of data classes, pattern matching on them (via the 'when' expression) is currently limited to matching the type. Consequently the balance() function is not very pretty!

// version 1.1.51

import Color.*

enum class Color { R, B }

sealed class Tree<A : Comparable<A>> {

    fun insert(x: A): Tree<A> {
        val t = ins(x)
        return when (t) {
            is T -> {
                val (_, a, y, b) = t
                T(B, a, y, b)
            }

            is E -> E()
        }
    }

    abstract fun ins(x: A): Tree<A>
}

class E<A : Comparable<A>> : Tree<A>() {

    override fun ins(x: A): Tree<A> = T(R, E(), x, E())

    override fun toString() = "E"
}

data class T<A : Comparable<A>>(
    val cl: Color,
    val le: Tree<A>,
    val aa: A,
    val ri: Tree<A>
) : Tree<A>() {

    private fun balance(): Tree<A> {
        if (cl != B) return this
        val res =
            if (le is T && le.le is T && le.cl == R && le.le.cl == R) {
               val (_, t, z, d) = this
               val (_, t2, y, c) = t as T
               val (_, a, x, b) = t2 as T
               T(R, T(B, a, x, b), y, T(B, c, z, d))
            }
            else if (le is T && le.ri is T && le.cl == R && le.ri.cl == R) {
               val (_, t, z, d) = this
               val (_, a, x, t2) = t as T
               val (_, b, y, c) = t2 as T
               T(R, T(B, a, x, b), y, T(B, c, z, d))
            }
            else if (ri is T && ri.le is T && ri.cl == R && ri.le.cl == R) {
               val (_, a, x, t) = this
               val (_, t2, z, d) = t as T
               val (_, b, y, c) = t2 as T
               T(R, T(B, a, x, b), y, T(B, c, z, d))
            }
            else if (ri is T && ri.ri is T && ri.cl == R && ri.ri.cl == R) {
               val (_, a, x, t) = this
               val (_, b, y, t2) = t as T
               val (_, c, z, d) = t2 as T
               T(R, T(B, a, x, b), y, T(B, c, z, d))
            }
            else this
        return res
    }

    override fun ins(x: A): Tree<A> = when (x.compareTo(aa)) {
        -1   -> T(cl, le.ins(x), aa, ri).balance()
        +1   -> T(cl, le, aa, ri.ins(x)).balance()
        else -> this
    }

    override fun toString() = "T($cl, $le, $aa, $ri)"
}

fun main(args: Array<String>) {
    var tree: Tree<Int> = E()
    for (i in 1..16) {
        tree = tree.insert(i)
    }
    println(tree)
}

Output:

T(B, T(B, T(B, T(B, E, 1, E), 2, T(B, E, 3, E)), 4, T(B, T(B, E, 5, E), 6, T(B, E, 7, E))), 8, T(B, T(B, T(B, E, 9, E), 10, T(B, E, 11, E)), 12, T(B, T(B, E, 13, E), 14, T(B, E, 15, T(R, E, 16, E)))))

Nim

Library: fusion/matching

import fusion/matching
{.experimental: "caseStmtMacros".}

type
  Colour = enum Empty, Red, Black
  RBTree[T] = ref object
    colour: Colour
    left, right: RBTree[T]
    value: T

proc `[]`[T](r: RBTree[T], idx: static[FieldIndex]): auto =
  ## enables tuple syntax for unpacking and matching
  when idx == 0: r.colour
  elif idx == 1: r.left
  elif idx == 2: r.value
  elif idx == 3: r.right

template B[T](l: untyped, v: T, r): RBTree[T] = 
  RBTree[T](colour: Black, left: l, value: v, right: r)

template R[T](l: untyped, v: T, r): RBTree[T] = 
  RBTree[T](colour: Red, left: l, value: v, right: r)

template balImpl[T](t: typed): untyped =
  case t
  of (colour: Red | Empty): discard
  of (Black, (Red, (Red, @a, @x, @b), @y, @c), @z, @d) |
    (Black, (Red, @a, @x, (Red, @b, @y, @c)), @z, @d) |
    (Black, @a, @x, (Red, (Red, @b, @y, @c), @z, @d)) |
    (Black, @a, @x, (Red, @b, @y, (Red, @c, @z, @d))):
    t = R(B(a, x, b), y, B(c, z, d))

proc balance*[T](t: var RBTree[T]) = balImpl[T](t)

template insImpl[T](t, x: typed): untyped =
  template E: RBTree[T] = RBTree[T]()
  case t
  of (colour: Empty): t = R(E, x, E)
  of (value: > x): t.left.ins(x); t.balance()
  of (value: < x): t.right.ins(x); t.balance()

proc insert*[T](tt: var RBTree[T], xx: T) =
  proc ins(t: var RBTree[T], x: T) = insImpl[T](t, x)
  tt.ins(xx)
  tt.colour = Black

OCaml

type color = R | B
type 'a tree = E | T of color * 'a tree * 'a * 'a tree

(** val balance : color * 'a tree * 'a * 'a tree -> 'a tree *)
let balance = function
  | B, T (R, T (R,a,x,b), y, c), z, d
  | B, T (R, a, x, T (R,b,y,c)), z, d
  | B, a, x, T (R, T (R,b,y,c), z, d)
  | B, a, x, T (R, b, y, T (R,c,z,d)) -> T (R, T (B,a,x,b), y, T (B,c,z,d))
  | col, a, x, b                      -> T (col, a, x, b) 

(** val insert : 'a -> 'a tree -> 'a tree *)
let insert x s = 
  let rec ins = function
    | E                  -> T (R,E,x,E)
    | T (col,a,y,b) as s ->
	if x < y then
	  balance (col, ins a, y, b)
	else if x > y then
	  balance (col, a, y, ins b)
	else
	  s
  in let T (_,a,y,b) = ins s 
  in T (B,a,y,b)

Oz

Translation of: Haskell

Unlike Haskell, Oz does not support multiple equations per function. So we use an explicit case-statement. To match multiple variables at once, we create temporary tuples with "#".

fun {Balance Col A X B}
   case Col#A#X#B
   of b#t(r t(r A X B) Y C         )#Z#D                            then t(r t(b A X B) Y t(b C Z D))
   [] b#t(r A          X t(r B Y C))#Z#D                            then t(r t(b A X B) Y t(b C Z D))
   [] b#A                           #X#t(r t(r B Y C) Z D)          then t(r t(b A X B) Y t(b C Z D))
   [] b#A                           #X#t(r B          Y t(r C Z D)) then t(r t(b A X B) Y t(b C Z D))
   else t(Col A X B)
   end
end

fun {Insert X S}
   fun {Ins S}
      case S of e then t(r e X e)
      [] t(Col A Y B) then
	 if X < Y then {Balance Col {Ins A} Y B}
	 elseif X > Y then {Balance Col A Y {Ins B}}
	 else S
	 end
      end
   end
   t(_ A Y B) = {Ins S}
in
   t(b A Y B)
end

Perl

Works with: Perl version 5.010

Although Perl does not have algebraic data types, it does have a wonderfully flexible regular expression engine, which is powerfully enough to perform the task.

However, representing a tree as a string, and repeatedly parsing that string, is truly inefficient way to solve the problem. Someday, someone will write a perl multi-method-dispatch module which is as amazing as Raku's, and then we can copy the Raku solution here.

The $balanced variable matches against either some data, or the empty tree (_), or, using perl's amazing recursive regular expression feature, a non-empty tree.

Each of the single letter variables declared right after $balanced, match an instance of $balanced, and if they succeed, store the result into the %+ hash.

#!perl
use 5.010;
use strict;
use warnings qw(FATAL all);

my $balanced = qr{([^<>,]++|<(?-1),(?-1),(?-1),(?-1)>)};
my ($a, $b, $c, $d, $x, $y, $z) = map +qr((?<$_>$balanced)),
	'a'..'d', 'x'..'z';
my $col = qr{(?<col>[RB])};

sub balance {
	local $_ = shift;
	if( /^<B,<R,<R,$a,$x,$b>,$y,$c>,$z,$d>\z/ or
		/^<B,<R,$a,$x,<R,$b,$y,$c>>,$z,$d>\z/ or
		/^<B,$a,$x,<R,<R,$b,$y,$c>,$z,$d>>\z/ or
		/^<B,$a,$x,<R,$b,$y,<R,$c,$z,$d>>>\z/ )
	{
		my ($aa, $bb, $cc, $dd) = @+{'a'..'d'};
		my ($xx, $yy, $zz) = @+{'x'..'z'};
		"<R,<B,$aa,$xx,$bb>,$yy,<B,$cc,$zz,$dd>>";
	} else {
		$_;
	}
}

sub ins {
	my ($xx, $tree) = @_;
	if($tree =~ m{^<$col,$a,$y,$b>\z} ) {
		my ($color, $aa, $bb, $yy) = @+{qw(col a b y)};
		if( $xx < $yy ) {
			return balance "<$color,".ins($xx,$aa).",$yy,$bb>";
		} elsif( $xx > $yy ) {
			return balance "<$color,$aa,$yy,".ins($xx,$bb).">";
		} else {
			return $tree;
		}
	} elsif( $tree !~ /,/) {
		return "<R,_,$xx,_>";
	} else {
		print "Unexpected failure!\n";
		print "Tree parts are: \n";
		print $_, "\n" for $tree =~ /$balanced/g;
		exit;
	}
}

sub insert {
	my $tree = ins(@_);
	$tree =~ m{^<$col,$a,$y,$b>\z} or die;
	"<B,$+{a},$+{y},$+{b}>";
}

MAIN: {
	my @a = 1..10;
	for my $aa ( 1 .. $#a ) {
		my $bb = int rand( 1 + $aa );
		@a[$aa, $bb] = @a[$bb, $aa];
	}
	my $t = "!";
	for( @a ) {
		$t = insert( $_, $t );
		print "Tree: $t.\n";
	}
}
print "Done\n";

Output:

Tree: <B,_,9,_>.
Tree: <B,<R,_,7,_>,9,_>.
Tree: <B,<B,_,2,_>,7,<B,_,9,_>>.
Tree: <B,<B,_,2,<R,_,6,_>>,7,<B,_,9,_>>.
Tree: <B,<B,_,2,<R,_,6,_>>,7,<B,_,9,<R,_,10,_>>>.
Tree: <B,<R,<B,_,2,_>,5,<B,_,6,_>>,7,<B,_,9,<R,_,10,_>>>.
Tree: <B,<R,<B,_,2,<R,_,4,_>>,5,<B,_,6,_>>,7,<B,_,9,<R,_,10,_>>>.
Tree: <B,<R,<B,_,2,<R,_,4,_>>,5,<B,_,6,_>>,7,<B,<R,_,8,_>,9,<R,_,10,_>>>.
Tree: <B,<R,<B,<R,_,1,_>,2,<R,_,4,_>>,5,<B,_,6,_>>,7,<B,<R,_,8,_>,9,<R,_,10,_>>>.
Tree: <B,<B,<B,<R,_,1,_>,2,_>,3,<B,_,4,_>>,5,<B,<B,_,6,_>,7,<B,<R,_,8,_>,9,<R,_,10,_>>>>.
Done

Phix

There is no formal support for this sort of thing in Phix, but that's not to say that whipping something up is likely to be particularly difficult, so let's give it a whirl.

--
-- demo\rosetta\Pattern_matching.exw
-- =================================
--
-- 1). Lightly modified copy of demo\rosetta\VisualiseTree.exw

with javascript_semantics

-- To the theme tune of the Milk Tray Ad iyrt,
-- All because the Windows console hates utf8:
constant TL = '\#DA', -- aka '┌'
         VT = '\#B3', -- aka '│'
         BL = '\#C0', -- aka '└'
         HZ = '\#C4', -- aka '─'
         HS = "\#C4"  -- (string version of HZ)

function w1252_to_utf8(string s)
    if platform()!=WINDOWS then
        s = substitute_all(s,{ TL, VT, BL, HZ},
                             {"┌","│","└","─"})
    end if
    return s
end function
--</hates utf8>

procedure visualise_tree(object tree, string root=HS)
    if atom(tree) then
        puts(1,"<empty>\n")
    else
        object {colour,left,v,right} = tree
        integer g = root[$]
        if sequence(left) then
            root[$] = iff(g=TL or g=HZ?' ':VT)
            visualise_tree(left,root&TL)
        end if
        root[$] = g
        printf(1,"%s%s%v\n",{w1252_to_utf8(root),colour,v})
        if sequence(right) then
            root[$] = iff(g=TL?VT:' ')
            visualise_tree(right,root&BL)
        end if
    end if
end procedure
--</copy VisualiseTree>

-- 2). Imagine the following is in a file, say algebraic_data_types.e - not quite generic enough
--      for inclusion in builtins, but not exactly difficult to copy/maintain per-project either.

function match_one(sequence key, object t)
    sequence res = {}
    if sequence(t)
    and length(key)==length(t) then
        for i=1 to length(key) do
            object ki = key[i], ti = t[i]
            if sequence(ki) and not string(ki) then
                sequence r2 = match_one(ki,ti)
                if r2={} then res = {} exit end if
                res &= r2
            else
                if ki=0 then
                    res = append(res,ti)
                else
                    if ki!=ti then res = {} exit end if
                end if
            end if
        end for
    end if
    return res
end function

/*global*/ function match_algebraic(sequence set, t)
    sequence s
    for i=1 to length(set) do
        s = match_one(set[i],t)
        if length(s) then exit end if
    end for
    return s
end function
--</algebraic_data_types.e>

-- 3). The actual task

constant B = "B", R = "R"

function balance(sequence t)
    sequence s = match_algebraic({{B,{R,{R,0,0,0},0,0},0,0},
                                  {B,{R,0,0,{R,0,0,0}},0,0},
                                  {B,0,0,{R,{R,0,0,0},0,0}},
                                  {B,0,0,{R,0,0,{R,0,0,0}}}},t)
    if length(s) then
        object {a,x,b,y,c,z,d} = s
        t = {R,{B,a,x,b},y,{B,c,z,d}}
    end if
    return t
end function

function ins(object tree, object leaf)
    if tree=NULL then
        tree = {R,NULL,leaf,NULL}
    else
        object {c,l,k,r} = tree
        if leaf!=k then
            if leaf<k then l = ins(l,leaf)
                      else r = ins(r,leaf)
            end if
            tree = balance({c,l,k,r})
        end if
    end if
    return tree
end function

function tree_insert(object tree, object leaf)
    tree = ins(tree,leaf)
    tree[1] = B
    return tree
end function

sequence stuff = shuffle(tagset(10))
object tree = NULL
for i=1 to length(stuff) do
    tree = tree_insert(tree,stuff[i])
end for
visualise_tree(tree)

?"done"
{} = wait_key()

Output:

   ┌R1
  ┌B2
 ┌B3
 │└B4
─B5
 │┌B6
 ││└R7
 └B8
  └B9
   └R10

Picat

Translation of: Prolog

main =>
    T = e,
    foreach (X in 1..10)
        insert(X,T,T1),
        T := T1
    end,
    output(T,0).

insert(X,S,R) =>
    ins(X,S,R1),
    R1 = $t(_,A,Y,B),
    R = $t(b,A,Y,B).

ins(X,e,R) => R = $t(r,e,X,e).
ins(X,t(C,A,Y,B),R), X < Y => ins(X,A,Ao), balance(C,Ao,Y,B,R).
ins(X,t(C,A,Y,B),R), X > Y => ins(X,B,Bo), balance(C,A,Y,Bo,R).
ins(_X,T,R) => R = T.

balance(C,A,X,B,S) :- (bal(C,A,X,B,T) -> S = T ; S = $t(C,A,X,B)).

bal(b, t(r,t(r,A,X,B),Y,C), Z, D, R) => R = $t(r,t(b,A,X,B),Y,t(b,C,Z,D)).
bal(b, t(r,A,X,t(r,B,Y,C)), Z, D, R) => R = $t(r,t(b,A,X,B),Y,t(b,C,Z,D)).
bal(b, A, X, t(r,t(r,B,Y,C),Z,D), R) => R = $t(r,t(b,A,X,B),Y,t(b,C,Z,D)).
bal(b, A, X, t(r,B,Y,t(r,C,Z,D)), R) => R = $t(r,t(b,A,X,B),Y,t(b,C,Z,D)).

output(e,Indent) => printf("%*w\n",Indent,e).
output(t(C,A,Y,B),Indent) =>
    output(A,Indent+6),
    printf("%*w[%w]\n",Indent,C,Y),
    output(B,Indent+6).

Output:

PicoLisp

Translation of: Prolog

(be color (R))
(be color (B))

(be tree (@ E))
(be tree (@P (T @C @L @X @R))
   (color @C)
   (tree @P @L)
   (call @P @X)
   (tree @P @R) )

(be bal (B (T R (T R @A @X @B) @Y @C) @Z @D (T R (T B @A @X @B) @Y (T B @C @Z @D))))
(be bal (B (T R @A @X (T R @B @Y @C)) @Z @D (T R (T B @A @X @B) @Y (T B @C @Z @D))))
(be bal (B @A @X (T R (T R @B @Y @C) @Z @D) (T R (T B @A @X @B) @Y (T B @C @Z @D))))
(be bal (B @A @X (T R @B @Y (T R @C @Z @D)) (T R (T B @A @X @B) @Y (T B @C @Z @D))))

(be balance (@C @A @X @B @S)
   (bal @C @A @X @B @S)
   T )
(be balance (@C @A @X @B (T @C @A @X @B)))

(be ins (@X E (T R E @X E)))
(be ins (@X (T @C @A @Y @B) @R)
   (^ @ (> (-> @Y) (-> @X)))
   (ins @X @A @Ao)
   (balance @C @Ao @Y @B @R)
   T )
(be ins (@X (T @C @A @Y @B) @R)
   (^ @ (> (-> @X) (-> @Y)))
   (ins @X @B @Bo)
   (balance @C @A @Y @Bo @R)
   T )
(be ins (@X (T @C @A @Y @B) (T @C @A @Y @B)))

(be insert (@X @S (T B @A @Y @B))
   (ins @X @S (T @ @A @Y @B)) )

Test:

: (? (insert 2 E @A) (insert 1 @A @B) (insert 3 @B @C))
 @A=(T B E 2 E) @B=(T B (T R E 1 E) 2 E) @C=(T B (T R E 1 E) 2 (T R E 3 E))
-> NIL

Prolog

color(r).
color(b).

tree(_,e).
tree(P,t(C,L,X,R)) :- color(C), tree(P,L), call(P,X), tree(P,R).

bal(b, t(r,t(r,A,X,B),Y,C), Z, D, t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, t(r,A,X,t(r,B,Y,C)), Z, D, t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, A, X, t(r,t(r,B,Y,C),Z,D), t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, A, X, t(r,B,Y,t(r,C,Z,D)), t(r,t(b,A,X,B),Y,t(b,C,Z,D))).

balance(C,A,X,B,S) :- ( bal(C,A,X,B,T) -> S = T ; S = t(C,A,X,B) ).

ins(X,e,t(r,e,X,e)).
ins(X,t(C,A,Y,B),R) :- ( X < Y -> ins(X,A,Ao), balance(C,Ao,Y,B,R)
                       ; X > Y -> ins(X,B,Bo), balance(C,A,Y,Bo,R)
                       ; X = Y -> R = t(C,A,Y,B)
                       ).

insert(X,S,t(b,A,Y,B)) :- ins(X,S,t(_,A,Y,B)).

Python

Translation of: C#

Structural pattern matching was added to Python in version 3.10.

from __future__ import annotations
from enum import Enum
from typing import NamedTuple
from typing import Optional


class Color(Enum):
    B = 0
    R = 1


class Tree(NamedTuple):
    color: Color
    left: Optional[Tree]
    value: int
    right: Optional[Tree]

    def insert(self, val: int) -> Tree:
        return self._insert(val).make_black()

    def _insert(self, val: int) -> Tree:
        match compare(val, self.value):
            case _ if self == EMPTY:
                return Tree(Color.R, EMPTY, val, EMPTY)
            case -1:
                assert self.left is not None
                return Tree(
                    self.color, self.left._insert(val), self.value, self.right
                ).balance()
            case 1:
                assert self.right is not None
                return Tree(
                    self.color, self.left, self.value, self.right._insert(val)
                ).balance()
            case _:
                return self

    def balance(self) -> Tree:
        match self:
            case (Color.B, (Color.R, (Color.R, a, x, b), y, c), z, d):
                return Tree(Color.R, Tree(Color.B, a, x, b), y, Tree(Color.B, c, z, d))
            case (Color.B, (Color.R, a, x, (Color.R, b, y, c)), z, d):
                return Tree(Color.R, Tree(Color.B, a, x, b), y, Tree(Color.B, c, z, d))
            case (Color.B, a, x, (Color.R, (Color.R, b, y, c), z, d)):
                return Tree(Color.R, Tree(Color.B, a, x, b), y, Tree(Color.B, c, z, d))
            case (Color.B, a, x, (Color.R, b, y, (Color.R, c, z, d))):
                return Tree(Color.R, Tree(Color.B, a, x, b), y, Tree(Color.B, c, z, d))
            case _:
                return self

    def make_black(self) -> Tree:
        return self._replace(color=Color.B)

    def __str__(self) -> str:
        if self == EMPTY:
            return "[]"
        return f"[{'R' if self.color == Color.R else 'B'}{self.value}]"

    def print(self, indent: int = 0) -> None:
        if self != EMPTY:
            assert self.right is not None
            self.right.print(indent + 1)

        print(f"{' ' * indent * 4}{self}")

        if self != EMPTY:
            assert self.left is not None
            self.left.print(indent + 1)


EMPTY = Tree(Color.B, None, 0, None)


def compare(x: int, y: int) -> int:
    if x > y:
        return 1
    if x < y:
        return -1
    return 0


def main():
    tree = EMPTY
    for i in range(1, 17):
        tree = tree.insert(i)
    tree.print()


if __name__ == "__main__":
    main()

Output:

Racket

Translation of: OCaml

#lang racket

;; Using short names to make the code line up nicely
(struct N (color left value right) #:prefab)

(define (balance t)
  (match t
    [(N 'B (N 'R (N 'R a x b) y c) z d) (N 'R (N 'B a x b) y (N 'B c z d))]
    [(N 'B (N 'R a x (N 'R b y c)) z d) (N 'R (N 'B a x b) y (N 'B c z d))]
    [(N 'B a x (N 'R (N 'R b y c) z d)) (N 'R (N 'B a x b) y (N 'B c z d))]
    [(N 'B a x (N 'R b y (N 'R c z d))) (N 'R (N 'B a x b) y (N 'B c z d))]
    [else t]))

(define (insert x s)
  (define (ins t)
    (match t
      ['empty (N 'R 'empty x 'empty)]
      [(N c l v r) (cond [(< x v) (balance (N c (ins l) v r))]
                         [(> x v) (balance (N c l v (ins r)))]
                         [else t])]))
  (match (ins s) [(N _ l v r) (N 'B l v r)]))

(define (visualize t0)
  (let loop ([t t0] [last? #t] [indent '()])
    (define (I mid last) (cond [(eq? t t0) ""] [last? mid] [else last]))
    (for-each display (reverse indent))
    (printf "~a~a[~a]\n" (I "\\-" "+-") (N-value t) (N-color t))
    (define subs (filter N? (list (N-left t) (N-right t))))
    (for ([s subs] [n (in-range (sub1 (length subs)) -1 -1)])
      (loop s (zero? n) (cons (I "  " "| ") indent)))))

(visualize (for/fold ([t 'empty]) ([i 16]) (insert i t)))

7[B]
+-3[B]
| +-1[B]
| | +-0[B]
| | \-2[B]
| \-5[B]
|   +-4[B]
|   \-6[B]
\-11[B]
  +-9[B]
  | +-8[B]
  | \-10[B]
  \-13[B]
    +-12[B]
    \-14[B]
      \-15[R]

Raku

(formerly Perl 6)

Works with: rakudo version 2016.11

Raku doesn't have algebraic data types (yet), but it does have pretty good pattern matching in multi signatures.

enum RedBlack <R B>;

multi balance(B,[R,[R,$a,$x,$b],$y,$c],$z,$d) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] }
multi balance(B,[R,$a,$x,[R,$b,$y,$c]],$z,$d) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] }
multi balance(B,$a,$x,[R,[R,$b,$y,$c],$z,$d]) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] }
multi balance(B,$a,$x,[R,$b,$y,[R,$c,$z,$d]]) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] }

multi balance($col, $a, $x, $b) { [$col, $a, $x, $b] }
 
multi ins( $x, @s [$col, $a, $y, $b] ) {
    when $x before $y     { balance $col, ins($x, $a), $y, $b }
    when $x after $y      { balance $col, $a, $y, ins($x, $b) }
    default               { @s }
}
multi ins( $x, Any:U ) { [R, Any, $x, Any] }

multi insert( $x, $s ) {
    [B, |ins($x,$s)[1..3]];
}

sub MAIN {
    my $t = Any;
    $t = insert($_, $t) for (1..10).pick(*);
    say $t.gist;
}

This code uses generic comparison operators before and after, so it should work on any ordered type.

Output:

[B [B [B (Any) 1 [R (Any) 2 (Any)]] 3 [B (Any) 4 [R (Any) 5 (Any)]]] 6 [B [B (Any) 7 (Any)] 8 [B [R (Any) 9 (Any)] 10 (Any)]]]

Rascal

Rascal offers many options for pattern matching. In essence, there are four sorts of patterns: Abstract, Concrete, PatternWithAction and classic Regular Expressions. These patterns can be used in several cases, for example switch or visit statements, on the right of the Match operator (:=), or in TryCatch statements for thrown exceptions. Each pattern binds variables in a conditional scope.

Abstract

An abstract pattern is recursively defined and may contain, among others, the following elements: Literal, VariableDeclaration, MultiVariable, Variable, List, Set, Tuple, Node, Descendant, Labelled, TypedLabelled, TypeConstrained. More explanation can be found in the Documentation. Some examples:

// Literal
rascal>123 := 123
bool: true

// VariableDeclaration
rascal>if(str S := "abc")
>>>>>>>   println("Match succeeds, S == \"<S>\"");
Match succeeds, S == "abc"
ok

// MultiVariable
rascal>if([10, N*, 50] := [10, 20, 30, 40, 50])
>>>>>>>   println("Match succeeds, N == <N>");
Match succeeds, N == [20,30,40]
ok

// Variable
rascal>N = 10;
int: 10
rascal>N := 10;
bool: true
rascal>N := 20;
bool: false

// Set and List
rascal>if({10, set[int] S, 50} := {50, 40, 30, 20, 10})
>>>>>>>   println("Match succeeded, S = <S>");
Match succeeded, S = {30,40,20}
ok

rascal>for([L1*, L2*] := [10, 20, 30, 40, 50]) 
>>>>>>>    println("<L1> and <L2>");
[] and [10,20,30,40,50]
[10] and [20,30,40,50]
[10,20] and [30,40,50]
[10,20,30] and [40,50]
[10,20,30,40] and [50]
[10,20,30,40,50] and []
list[void]: []

// Descendant
rascal>T = red(red(black(leaf(1), leaf(2)), black(leaf(3), leaf(4))), black(leaf(5), leaf(4)));
rascal>for(/black(_,leaf(4)) := T)
>>>>>>>    println("Match!");
Match!
Match!
list[void]: []

rascal>for(/black(_,leaf(int N)) := T)
>>>>>>>    println("Match <N>");
Match 2
Match 4
Match 4
list[void]: []

rascal>for(/int N := T)
>>>>>>>    append N;
list[int]: [1,2,3,4,5,4]

// Labelled
rascal>for(/M:black(_,leaf(4)) := T)
>>>>>>>    println("Match <M>");
Match black(leaf(3),leaf(4))
Match black(leaf(5),leaf(4))
list[void]: []

Concrete

Suppose we want to manipulate text written in some hypothetical language LANG. Then first the concrete syntax of LANG has to be defined by importing a module that declares the non-terminals and syntax rules for LANG. Next LANG programs have to be parsed. LANG programs made come from text files or they may be included in the Rascal program as literals. In both cases the text is parsed according to the defined syntax and the result is a parse tree in the form of a value of type Tree. Concrete patterns operate on these trees.

A concrete pattern is a quoted concrete syntax fragment that may contain variables. The syntax that is used to parse the concrete pattern may come from any module that has been imported in the module in which the concrete pattern occurs. Some examples of concrete patterns:

// Quoted pattern 
` Token1 Token2 ... Tokenn `
// A typed quoted pattern 
(Symbol) ` Token1 Token2 ... TokenN `
// A typed variable pattern 
<Type Var>
// A variable pattern 
<Var>

A full example of concrete patterns can be found in the Rascal Recipes.

PatternWithAction

There are two variants of the PatternsWitchAction. When the subject matches Pattern, the expression Exp is evaluated and the subject is replaced with the result. Secondly, when the subject matches Pattern, the (block of) Statement(s) is executed. See below for some ColoredTree examples:

// Define ColoredTrees with red and black nodes and integer leaves
data ColoredTree = leaf(int N)      
                 | red(ColoredTree left, ColoredTree right) 
                 | black(ColoredTree left, ColoredTree right);
          
// Count the number of black nodes        
public int cntBlack(ColoredTree t){
   int c = 0;
   visit(t) {
     case black(_,_): c += 1;      
   };
   return c;
}

// Returns if a tree is balanced
public bool balance(ColoredTree t){
   visit(t){
     case black(a,b): if (cntBlack(a) == cntBlack(b)) true; else return false;
     case red(a,b): if (cntBlack(a) == cntBlack(b)) true; else return false;
     }
   return true;
}
// Compute the sum of all integer leaves
public int addLeaves(ColoredTree t){
   int c = 0;
   visit(t) {
     case leaf(int N): c += N;   
   };
   return c;
}

// Add green nodes to ColoredTree
data ColoredTree = green(ColoredTree left, ColoredTree right); 

// Transform red nodes into green nodes
public ColoredTree makeGreen(ColoredTree t){
   return visit(t) {
     case red(l, r) => green(l, r)   
   };
}

Regular Expressions

Regular expressions are noated between two slashes. Most normal regular expressions patterns are available, such as ., \n, \d, etc. Additionally, flags can be used to create case intensiveness.

rascal>/XX/i := "some xx";
bool: true
rascal>/a.c/ := "abc";
bool: true

REXX

The nodes used for this example are taken from the Wikipedia example at: [red black tree, an example]

/*REXX pgm builds a red/black tree (with verification & validation), balanced as needed.*/
parse arg nodes '/' insert                       /*obtain optional arguments from the CL*/
if  nodes=''  then nodes =  13.8.17  8.1.11  17.15.25  1.6  25.22.27   /*default nodes. */
if insert=''  then insert=  22.44    44.66                             /*   "   inserts.*/
top= .                                           /*define the default for the  TOP  var.*/
call Dnodes nodes                                /*define nodes, balance them as added. */
call Dnodes insert                               /*insert   "       "      "   " needed.*/
call Lnodes                                      /*list the nodes  (with indentations). */
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
err:    say;              say '***error***: '   arg(1);         say;              exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
Dnodes: arg $d;   do j=1  for words($d);   t= word($d, j)     /*color: encoded into L.  */
                  parse var  t   p   '.'   a   "."   b   '.'   x   1  .  .  .  xx
                  call Vnodes p a b
                  if x\==''   then call err "too many nodes specified: "   xx
                  if p\==top  then if @.p==.  then call err "node isn't defined: "  p
                  if p ==top  then do;  !.p=1;  L.1=p;  end   /*assign the top node.    */
                  @.p= a b;   n= !.p + 1                      /*assign node; bump level.*/
                  if a\==''   then do;  !.a= n;    @.a=;    maxL= max(maxL, !.a);    end
                  if b\==''   then do;  !.b= n;    @.b=;    maxL= max(maxL, !.b);    end
                  L.n= space(L.n a b)                         /*append to the level list*/
                  end   /*j*/
        return
/*──────────────────────────────────────────────────────────────────────────────────────*/
Lnodes:           do L=1  for  maxL;  w= length(maxL);  rb= word('(red) (black)', 1+L//2)
                  say "level:"   right(L, w)   left('', L+L)   " ───► "   rb    ' '   L.L
                  end   /*lev*/
        return
/*──────────────────────────────────────────────────────────────────────────────────────*/
Vnodes: arg $v;   do v=1  for words($v);      y= word($v, v)
                  if \datatype(y, 'W')   then call err "node isn't a whole number: "   y
                  y= y / 1                                /*normalize Y int.: no LZ, dot*/
                  if top==.  then do;  LO=y;  top=y;    HI=y;   L.=;   @.=;  maxL=1;   end
                                       LO= min(LO, y);  HI= max(HI, y)
                  if @.y\==.  &  @.y\==''  then call err "node is already defined: "   y
                  end   /*v*/
        return

output when using the default inputs:

level: 1     ───►  (black)   13
level: 2       ───►  (red)   8 17
level: 3         ───►  (black)   1 11 15 25
level: 4           ───►  (red)   6 22 27
level: 5             ───►  (black)   44
level: 6               ───►  (red)   66

Rust

Translation of: Haskell

This would be a horribly inefficient way to implement a Red-Black Tree in Rust as nodes are being allocated and deallocated constantly, but it does show off Rust's pattern matching.

#![feature(box_patterns, box_syntax)]
use self::Color::*;
use std::cmp::Ordering::*;

enum Color {R,B}

type Link<T> = Option<Box<N<T>>>;
struct N<T> {
    c: Color,
    l: Link<T>,
    val: T,
    r: Link<T>,
}


impl<T: Ord> N<T> {
    fn balance(col: Color, n1: Link<T>, z: T, n2: Link<T>) -> Link<T> {
        Some(box 
             match (col,n1,n2) {
                   (B, Some(box N {c: R, l: Some(box N {c: R, l: a, val: x, r: b}), val: y, r: c}), d)
                |  (B, Some(box N {c: R, l: a, val: x, r: Some (box N {c: R, l: b, val: y, r: c})}), d)
                => N {c: R, l: Some(box N {c: B, l: a, val: x, r: b}), val: y, r: Some(box N {c: B, l: c, val: z, r: d})},
                   (B, a, Some(box N {c: R, l: Some(box N {c: R, l: b, val: y, r: c}), val: v, r: d}))
                |  (B, a, Some(box N {c: R, l: b, val: y, r: Some(box N {c: R, l: c, val: v, r: d})}))
                => N {c: R, l: Some(box N {c: B, l: a, val: z, r: b}), val: y, r: Some(box N {c: B, l: c, val: v, r: d})},
                (col, a, b) => N {c: col, l: a, val: z, r: b},
        })
    }
    fn insert(x: T, n: Link<T>) -> Link<T> {
        match n {
            None => Some(box N { c: R, l: None, val: x, r: None }),
            Some(n) =>  {
                let n = *n;
                let N {c: col, l: a, val: y, r: b} = n;
                match x.cmp(&y) {
                    Greater => Self::balance(col, a,y,Self::insert(x,b)),
                    Less    => Self::balance(col, Self::insert(x,a),y,b),
                    Equal   => Some(box N {c: col, l: a, val: y, r: b})
                }
            }
        }
    }
}

Scala

Translation of: Haskell

Algebraic data types are implemented in Scala through the combination of a number of different features, to ensure principles of Object Oriented Programming.

The main type is usually defined as a sealed abstract class, which ensures it can't be instantiated, and guarantees that it can't be expanded outside the file it was defined at. This last feature is used so the compiler can verify that the pattern matching is complete, or warn when there are missing cases. It can be ommitted if preferred.

Each subtype is defined either as a case object, for non-paremeterized types, or case class, for parameterized types, all extending the main type. The case keyword is not required, but, when used, it provides a number of default methods which ensure they can be used without any further definitions.

The most important of those default methods for the purpose of algebraic data types is the extractor, a method called either unapply or unapplySeq, and which returns an Option containing the deconstructed parameters, or None if the passed object can't be deconstructed by this method. Scala uses the extractors to implement pattern matching without exposing the internal representation of the data.

This specific task is made much harder than necessary because Scala doesn't have a variant ordering class. Given that limitation, one has to either give up on a singleton object representing the empty tree, or give up on parameterizing the tree itself.

The solution below, uses the latter approach. The algebraic data types are members of a RedBlackTree class, which, itself, receives a type parameter for the keys of the tree, and an implicit parameter for an Ordering for that type. To use the tree it is thus necessary to instantiate an object of type RedBlackTree, and then reference the members of that object.

class RedBlackTree[A](implicit ord: Ordering[A]) {
  sealed abstract class Color
  case object R extends Color
  case object B extends Color
  
  sealed abstract class Tree {
    def insert(x: A): Tree = ins(x) match {
      case T(_, a, y, b) => T(B, a, y, b)
      case E             => E
    }
    def ins(x: A): Tree
  }
  
  case object E extends Tree {
    override def ins(x: A): Tree = T(R, E, x, E) 
  }
  
  case class T(c: Color, left: Tree, a: A, right: Tree) extends Tree {
    private def balance: Tree = (c, left, a, right) match {
      case (B, T(R, T(R, a, x, b), y, c),             z, d                                    ) => T(R, T(B, a, x, b), y, T(B, c, z, d))
      case (B, T(R, a,             x, T(R, b, y, c)), z, d                                    ) => T(R, T(B, a, x, b), y, T(B, c, z, d))
      case (B, a,                                     x, T(R, T(R, b, y, c), z, d            )) => T(R, T(B, a, x, b), y, T(B, c, z, d))
      case (B, a,                                     x, T(R, b,             y, T(R, c, z, d))) => T(R, T(B, a, x, b), y, T(B, c, z, d))
      case _ => this
    }
    
    override def ins(x: A): Tree = ord.compare(x, a) match {
      case -1 => T(c, left ins x, a, right      ).balance
      case  1 => T(c, left,       a, right ins x).balance
      case  0 => this
    }
  }
}

Usage example:

scala> val rbt = new RedBlackTree[Int]
rbt: RedBlackTree[Int] = RedBlackTree@17dfcf1

scala> import rbt._
import rbt._

scala> List.range(1, 17).foldLeft(E: Tree)(_ insert _)
res5: rbt.Tree = T(B,T(B,T(B,T(B,E,1,E),2,T(B,E,3,E)),4,T(B,T(B,E,5,E),6,T(B,E,7,E))),8,T(B,T(B,T(B,E,9,E),10,T(B,E,11,E
)),12,T(B,T(B,E,13,E),14,T(B,E,15,T(R,E,16,E)))))

Standard ML

datatype color = R | B
datatype 'a tree = E | T of color * 'a tree * 'a * 'a tree

(** val balance = fn : color * 'a tree * 'a * 'a tree -> 'a tree *)
fun balance (B, T (R, T (R,a,x,b), y, c), z, d) = T (R, T (B,a,x,b), y, T (B,c,z,d))
  | balance (B, T (R, a, x, T (R,b,y,c)), z, d) = T (R, T (B,a,x,b), y, T (B,c,z,d))
  | balance (B, a, x, T (R, T (R,b,y,c), z, d)) = T (R, T (B,a,x,b), y, T (B,c,z,d))
  | balance (B, a, x, T (R, b, y, T (R,c,z,d))) = T (R, T (B,a,x,b), y, T (B,c,z,d))
  | balance (col, a, x, b)                      = T (col, a, x, b) 

(** val insert = fn : int -> int tree -> int tree *)
fun insert x s = let
  fun ins E                    = T (R,E,x,E)
    | ins (s as T (col,a,y,b)) =
	if x < y then
	  balance (col, ins a, y, b)
	else if x > y then
	  balance (col, a, y, ins b)
	else
	  s
  val T (_,a,y,b) = ins s 
in
  T (B,a,y,b)
end

Swift

Works with: Swift version 2+

enum Color { case R, B }
enum Tree<A> {
  case E
  indirect case T(Color, Tree<A>, A, Tree<A>)
}

func balance<A>(input: (Color, Tree<A>, A, Tree<A>)) -> Tree<A> {
  switch input {
  case let (.B, .T(.R, .T(.R,a,x,b), y, c), z, d): return .T(.R, .T(.B,a,x,b), y, .T(.B,c,z,d))
  case let (.B, .T(.R, a, x, .T(.R,b,y,c)), z, d): return .T(.R, .T(.B,a,x,b), y, .T(.B,c,z,d))
  case let (.B, a, x, .T(.R, .T(.R,b,y,c), z, d)): return .T(.R, .T(.B,a,x,b), y, .T(.B,c,z,d))
  case let (.B, a, x, .T(.R, b, y, .T(.R,c,z,d))): return .T(.R, .T(.B,a,x,b), y, .T(.B,c,z,d))
  case let (col, a, x, b)                        : return .T(col, a, x, b)
  }
}

func insert<A : Comparable>(x: A, s: Tree<A>) -> Tree<A> {
  func ins(s: Tree<A>) -> Tree<A> {
    switch s {
    case     .E           : return .T(.R,.E,x,.E)
    case let .T(col,a,y,b):
      if x < y {
        return balance((col, ins(a), y, b))
      } else if x > y {
        return balance((col, a, y, ins(b)))
      } else {
        return s
      }
    }
  }
  switch ins(s) {
  case let .T(_,a,y,b): return .T(.B,a,y,b)
  case     .E:
    assert(false)
    return .E
  }
}

Tailspin

Translation of: Haskell

Tailspin doesn't have type names, so here using a tag. Neither does it have destructuring (which seems to be posited in the problem statement). Arguably, pattern matching in Tailspin is more readable while still as concise.

processor RedBlackTree
  data node <{VOID}|{colour: <='black'|='red'>, left: <node>, right: <node>, value: <> VOID}> local
  @: {};
  sink insert
    templates balance
      when <{colour: <='black'>, left: <{ colour: <='red'> left: <{colour: <='red'>}>}>}>
        do { colour: 'red',
          left: { $.left.left..., colour: 'black'},
          value: $.left.value,
          right: { $..., left: $.left.right }} !
      when <{colour: <='black'>, left: <{ colour: <='red'> right: <{colour: <='red'>}>}>}>
        do { colour: 'red',
          left: { $.left..., colour: 'black', right: $.left.right.left},
          value: $.left.right.value,
          right: { $..., left: $.left.right.right }} !
      when <{colour: <='black'>, right: <{ colour: <='red'> left: <{colour: <='red'>}>}>}>
        do { colour: 'red',
          left: { $..., right: $.right.left.left},
          value: $.right.left.value,
          right: { $.right..., colour: 'black', left: $.right.left.right }} !
      when <{colour: <='black'>, right: <{ colour: <='red'> right: <{colour: <='red'>}>}>}>
        do { colour: 'red',
          left: { $..., right: $.right.left},
          value: $.right.value,
          right: { $.right.right..., colour: 'black' }} !
      otherwise $ !
    end balance
    templates ins&{into:}
      when <?($into <´node´ ={}>)> do { colour: 'red', left: {}, value: $, right: {}} !
      when <..$into.value::raw> do { $into..., left: $ -> ins&{into: $into.left}} -> balance !
      otherwise { $into..., right: $ -> ins&{into: $into.right}} -> balance !
    end ins
    @RedBlackTree: { $ -> ins&{into: $@RedBlackTree} ..., colour: 'black'};
  end insert
  source toString
    '$@RedBlackTree;' !
  end toString
end RedBlackTree

def tree: $RedBlackTree;
1..5 -> \('$tree::toString;$#10;' -> !OUT::write $ -> !tree::insert \) -> !VOID
$tree::toString -> !OUT::write

Output:

{}
{colour=black, left={}, right={}, value=1}
{colour=black, left={}, right={colour=red, left={}, right={}, value=2}, value=1}
{colour=black, left={colour=black, left={}, right={}, value=1}, right={colour=black, left={}, right={}, value=3}, value=2}
{colour=black, left={colour=black, left={}, right={}, value=1}, right={colour=black, left={}, right={colour=red, left={}, right={}, value=4}, value=3}, value=2}
{colour=black, left={colour=black, left={}, right={}, value=1}, right={colour=red, left={colour=black, left={}, right={}, value=3}, right={colour=black, left={}, right={}, value=5}, value=4}, value=2}

Tcl

Translation of: Haskell

Tcl doesn't have algebraic types built-in, but they can be simulated using tagged lists, and a custom pattern matching control structure can be built:

# From http://wiki.tcl.tk/9547
package require Tcl         8.5
package provide datatype    0.1

namespace eval ::datatype {
    namespace export define match matches
    namespace ensemble create

    # Datatype definitions
    proc define {type = args} { 
        set ns [uplevel 1 { namespace current }]
        foreach cons [split [join $args] |] {
            set name [lindex $cons 0]
            set args [lrange $cons 1 end]
            proc $ns\::$name $args [format {
                lreplace [info level 0] 0 0 %s
            } [list $name]]
        }
        return $type
    }

    # Pattern matching
    # matches pattern value envVar --
    #   Returns 1 if value matches pattern, else 0
    #   Binds match variables in envVar
    proc matches {pattern value envVar} {
        upvar 1 $envVar env
        if {[var? $pattern]} { return [bind env $pattern $value] }
        if {[llength $pattern] != [llength $value]} { return 0 }
        if {[lindex $pattern 0] ne [lindex $value 0]} { return 0 }
        foreach pat [lrange $pattern 1 end] val [lrange $value 1 end] {
            if {![matches $pat $val env]} { return 0 }
        }
        return 1
    }
    # A variable starts with lower-case letter or _. _ is a wildcard.
    proc var? term { string match {[a-z_]*} $term }
    proc bind {envVar var value} {
        upvar 1 $envVar env
        if {![info exists env]} { set env [dict create] }
        if {$var eq "_"} { return 1 }
        dict set env $var $value
        return 1
    }
    proc match args {
        #puts "MATCH: $args"
        set values [lrange $args 0 end-1]
        set choices [lindex $args end]
        append choices \n [list return -code error -level 2 "no match for $values"]
        set f [list values $choices [namespace current]]
        lassign [apply $f $values] env body
        #puts "RESULT: $env -> $body"
        dict for {k v} $env { upvar 1 $k var; set var $v }
        catch { uplevel 1 $body } msg opts
        dict incr opts -level
        return -options $opts $msg
    }
    proc case args {
        upvar 1 values values
        set patterns [lrange $args 0 end-2]
        set body [lindex $args end]
        set env [dict create]
        if {[llength $patterns] != [llength $values]} { return }
        foreach pattern $patterns value $values {
            if {![matches $pattern $value env]} { return }
        }
        return -code return [list $env $body]
    }
    proc default body { return -code return [list {} $body] }
}

We can then code our solution similar to Haskell:

datatype define Color = R | B
datatype define Tree  = E | T color left val right

# balance :: Color -> Tree a -> a -> Tree a -> Tree a
proc balance {color left val right} {
    datatype match $color $left $val $right {
        case B [T R [T R a x b] y c] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B [T R a x [T R b y c]] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B a x [T R [T R b y c] z d] -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case B a x [T R b y [T R c z d]] -> { T R [T B $a $x $b] $y [T B $c $z $d] }
        case col a x b                   -> { T $col $a $x $b }
    }
}
# insert :: Ord a => a -> Tree a -> Tree a
proc insert {x s} {
    datatype match [ins $x $s] {
        case [T _ a y b]  -> { T B $a $y $b }
    }
}
# ins :: Ord a => a -> Tree a -> Tree a
proc ins {x s} {
    datatype match $s {
        case E               -> { T R E $x E }
        case [T col a y b]   -> {
            if {$x < $y} { return [balance $col [ins $x $a] $y $b] }
            if {$x > $y} { return [balance $col $a $y [ins $x $b]] }
            return $s
        }
    }
}

TXR

TXR Lisp has structural pattern matching on objects of all kinds, including structures. We define a red-black tree structure like this, with a BOA constructor (by-order of arguments) for convenience:

(defstruct (rbnode color left right data) ()
  color
  left
  right
  data)

The empty tree case is handled by the nil symbol, so in terms of algebraic types, the tree is a sum of nil and the rbnode struct type, and that struct type is a product type of several properties. For the color slot, we use the keyword symbols :red and :black which needs not be declared anywhere. data can be any value.

TXR Lisp's syntax for matching structures looks like this:

@(struct time year @y month @m)

This example matches a time structure instance, capturing the year as y and month as m.

Structures aren't ordered tuples; they are clumps of of named slots, that cannot be accessed by position. This would break under inheritance, in particular multiple inheritance.

Furthermore, variables have the @ sigil in most pattern matching constructs, because symbols without the sigil denote themselves as literal patterns. The pattern x matches the symbol x literally, and no other object. The pattern @x matches any object and captures it as x.

These above features make it verbose and somewhat noisy to express pattern matching of our rbtree node. However, TXR Lisp's pattern matching sublanguage supports application-defined macro patterns, defined by the defmatch macro. With these we can achieve a shorthand notation which matches nodes as if they were ordered tuples, and which drops the sigils from variables.

(defmatch rb (color left right data)
  (flet ((var? (sym) (if (bindable sym) ^@,sym sym)))
    ^@(struct rbnode
        color ,(var? color)
        left ,(var? left)
        right ,(var? right)
        data ,(var? data))))

(defmatch red (left right data)
  ^@(rb :red ,left ,right ,data))

(defmatch black (left right data)
  ^@(rb :black ,left ,right ,data))

And with all the above, we can write the code like this:

(defun-match rb-balance
  ((@(or @(black @(red @(red a b x) c y) d z)
         @(black @(red a @(red b c x) x) d z)
         @(black a @(red @(red b c y) d z) x)
         @(black a @(red b @(red c d z) y) x)))
   (new (rbnode :red
                (new (rbnode :black a b x))
                (new (rbnode :black c d z))
                y)))
  ((@else) else))

(defun rb-insert-rec (tree x)
  (match-ecase tree
    (nil
     (new (rbnode :red nil nil x)))
    (@(rb color a b y)
     (cond
       ((< x y)
        (rb-balance (new (rbnode color (rb-insert-rec a) b y))))
       ((> x y)
        (rb-balance (new (rbnode color a (rb-insert-rec b) y))))
       (t tree)))))

(defun rb-insert (tree x)
  (match-case (rb-insert-rec tree x)
    (@(red a b y) (new (rbnode :black a b y)))
    (@else else)))

Insertion is split into two functions: a recursive one which works on its own, except that whenever the tree ends up with a red root, we would like to rewrite that node to a black one. We make the insertion function call the recursive one and then do this fix-up using pattern matching again.

Wren

Translation of: Go

Wren doesn't have either algebraic data types or pattern matching though, despite that, the T.balance() method looks better than I thought it would :)

var R = "R"
var B = "B"

class Tree {
    ins(x) {}    // overridden by child classes

    insert(x) {  // inherited by child classes
        var t = ins(x)
        if (t.type == T) return T.new(B, t.le, t.aa, t.ri)
        if (t.type == E) return E.new()
        return null
    } 
}

class T is Tree {
    construct new(cl, le, aa, ri) {
        _cl = cl  // color
        _le = le  // Tree
        _aa = aa  // integer
        _ri = ri  // Tree
    }

    cl { _cl }  
    le { _le }
    aa { _aa }
    ri { _ri }

    balance() {
        if (_cl != B) return this

        var le2 = _le.type == T ? _le : null
        var lele
        var leri
        if (le2) {
            lele = _le.le.type == T ? _le.le : null
            leri = _le.ri.type == T ? _le.ri : null
        }
        var ri2 = _ri.type == T ? _ri : null
        var rile
        var riri
        if (ri2) {
            rile = _ri.le.type == T ? _ri.le : null
            riri = _ri.ri.type == T ? _ri.ri : null
        }

        if (le2 && lele && le2.cl == R && lele.cl == R) {
            var t = le2.le
            return T.new(R, T.new(B, t.le, t.aa, t.ri), le2.aa, T.new(B, le2.ri, _aa, _ri))  
        }
        if (le2 && leri && le2.cl == R && leri.cl == R) {
            var t = le2.ri
            return T.new(R, T.new(B, le2.le, le2.aa, t.le), t.aa, T.new(B, t.ri, _aa, _ri))
        }
        if (ri2 && rile && ri2.cl == R && rile.cl == R) {
            var t = ri2.ri
            return T.new(R, T.new(B, _le, _aa, t.le), t.aa, T.new(B, t.ri, ri2.aa, ri2.ri))
        }
        if (ri2 && riri && ri2.cl == R && riri.cl == R) {
            var t = ri2.ri
            return T.new(R, T.new(B, _le, _aa, ri2.le), ri2.aa, T.new(B, t.le, t.aa, t.ri))
        }
        return this
    }

    ins(x) {
        if (x < _aa) return T.new(_cl, _le.ins(x), _aa, _ri).balance()
        if (x > _aa) return T.new(_cl, _le, _aa, _ri.ins(x)).balance()
        return this
    }

    toString { "T(%(_cl), %(_le), %(_aa), %(_ri))" }
}

class E is Tree {
    construct new() {}

    ins(x) { T.new(R, E.new(), x, E.new()) }

    toString { "E" }
}

var tr = E.new()
for (i in 1..16) tr = tr.insert(i)
System.print(tr)

Output:

T(B, T(B, T(B, T(B, E, 1, E), 2, T(B, E, 3, E)), 4, T(B, T(B, E, 5, E), 6, T(B, E, 7, E))), 8, T(B, T(B, T(B, E, 9, E), 10, T(B, E, 11, E)), 12, T(B, T(B, E, 13, E), 14, T(B, E, 15, T(R, E, 16, E)))))