Red black tree sort
Implement red-black tree sorting of fixed width integers. Here, the left branch will only contain nodes with a smaller key and the right branch will only contain nodes with a larger key.
Start with an empty tree, add 30 nodes each with arbitrary (aka "random") keys, then traverse the tree, printing the values from the left node, the key value, then the values from the right node, displaying their value and their color (red or black). Since we are using a red-black tree here, this would eliminate any duplicate values.
Then delete an arbitrary 15 nodes and display the resulting tree.
You may use an implementation at Algebraic data types as a starting point, if you find that helpful.
F#
The Functions
This task extends Algebraic data types#F.23 adding the following functions: <lang fsharp> // Red black tree sort. Nigel Galloway: June 17th., 2022 let fromSeq n=n|>Seq.fold(fun n g->insert g n) Empty let toSeq g=let rec fN g=seq{match g with N(i,g,e,l)->yield l; yield! fN g; yield! fN e |_->()} in fN g let delSeq n g=toSeq g|>Seq.except n|>fromSeq let rec printN n s t=match n with N(i,g,e,l)->printN g (s+" ") "L";printfn "%s %s %A %d" s t i l; printN e (s+" ") "R" |_->() </lang>
The Task
<lang fsharp> let n=fromSeq [85; 57; 51; 50; 86; 39; 95; 49; 44; 48; 21; 83; 11; 59; 88; 66; 4; 40; 24; 82; 63; 22; 37; 32; 91; 74; 28; 75; 62; 81] printfn "Populated Red Black Tree"; printN n "" "X" let g=delSeq [32; 40; 57; 63; 66; 75; 86; 59; 83; 51; 24; 62; 82; 39; 37] n printfn "\nRed Black Tree after deletions"; printN g "" "X" </lang>
- Output:
Populated Red Black Tree L Red 4 L Black 11 L Black 21 R Black 22 L Red 24 L Red 28 L Black 32 L Red 37 R Black 39 R Red 40 R Black 44 L Red 48 R Black 49 L Black 50 L Black 51 R Black 57 L Black 59 R Red 62 R Black 63 X Black 66 L Black 74 L Black 75 L Red 81 R Black 82 R Black 83 L Black 85 R Black 86 L Black 88 R Red 91 R Black 95 Red Black Tree after deletions L Red 4 L Black 11 L Black 21 R Black 22 R Red 28 X Black 44 L Black 48 L Black 49 R Black 50 R Red 74 L Black 81 R Black 85 L Red 88 R Black 91 R Red 95
FreeBASIC
Code originally by AGS. https://www.freebasic.net/forum/viewtopic.php?t=16113 <lang freebasic>#define NULL Cast(Any Ptr,0)
Enum nodecolor
BLACK = 0 RED = 1
End Enum
Type RBNode
Dim izda As RBNode Ptr Dim dcha As RBNode Ptr Dim parent As RBNode Ptr Dim kolor As nodecolor Dim key As Integer Dim value As String Dim nonzero As Integer Declare Constructor(Byval key As Integer = 0, value As String = "", Byval clr As nodecolor = RED) Declare Destructor()
End Type
Constructor RBNode(Byval key As Integer, value As String, Byval clr As nodecolor = RED)
This.key = key This.value = value This.izda = NULL This.dcha = NULL This.parent = NULL This.kolor = clr This.nonzero = 1
End Constructor
Destructor RBNode()
End Destructor
Function integer_compare(Byval key1 As Integer, Byval key2 As Integer) As Integer
If (key1 = key2) Then Return 0 Elseif (key1 < key2) Then Return -1 Elseif (key1 > key2) Then Return 1 End If
End Function
Type RBTree
Dim sentinel As RBNode Ptr Dim root As RBNode Ptr Dim count As Integer Dim Compare As Function(Byval key1 As Integer, Byval key2 As Integer) As Integer Declare Constructor(Byval cmp As Function(Byval key1 As Integer, Byval key2 As Integer) As Integer) Declare Sub rotateLeft(Byval x As RBNode Ptr) Declare Sub rotateRight(Byval x As RBNode Ptr) Declare Sub insertFixup(Byval x As RBNode Ptr) Declare Function insertNode(Byval key As Integer, value As String) As RBNode Ptr Declare Sub deleteFixup(Byval x As RBNode Ptr) Declare Sub deleteNode(Byval z As RBNode Ptr) Declare Function findNode(Byval key As Integer) As RBNode Ptr Declare Destructor()
End Type
Constructor RBTree(Byval cmp As Function(Byval key1 As Integer, Byval key2 As Integer) As Integer)
This.sentinel = New RBNode(0,"",BLACK) This.sentinel->izda = sentinel This.sentinel->dcha = sentinel This.root = This.sentinel This.count = 0 This.Compare = cmp
End Constructor
Destructor RBTree()
'The tree is transformed into a tree in which 'left children are always leaves. This is done by rotation. 'After rotating any left child is a leaf (not a tree) 'so a izda child can simply be deleted. 'Usually a stack is used to keep track of what nodes have 'been removed. By using rotation there is no need for a stack.
Dim parent As RBNode Ptr Dim child As RBNode Ptr If (This.root <> This.sentinel Andalso This.root <> NULL) Then parent = This.root While (parent <> This.sentinel) If (parent->izda = This.sentinel) Then child = parent->dcha Delete parent parent = 0 Else 'rotate child = parent->izda parent->izda = child->dcha child->dcha = parent End If parent = child Wend Else If (This.sentinel <> 0) Then Delete This.sentinel This.sentinel = 0 End If End If
End Destructor
Sub RBTree.rotateLeft(Byval x As RBNode Ptr)
'rotate node x to right Var y = x->dcha 'establish x->dcha link x->dcha = y->izda If (y->izda <> This.sentinel) Then y->izda->parent = x End If 'establish y->parent link If (y <> This.sentinel) Then y->parent = x->parent End If If (x->parent) Then If (x = x->parent->izda) Then x->parent->izda = y Else x->parent->dcha = y End If Else This.root = y End If 'link x and y y->izda = x If (x <> This.sentinel) Then x->parent = y End If
End Sub
Sub RBTree.rotateRight(Byval x As RBNode Ptr)
'rotate node x to right Var y = x->izda ' establish x->left link x->izda = y->dcha If (y->dcha <> This.sentinel) Then y->dcha->parent = x End If ' establish y->parent link If (y <> This.sentinel) Then y->parent = x->parent End If If (x->parent) Then If (x = x->parent->dcha) Then x->parent->dcha = y Else x->parent->izda = y End If Else This.root = y End If 'link x and y y->dcha = x If (x <> This.sentinel) Then x->parent = y End If
End Sub
Sub RBTree.insertFixup(Byval x As RBNode Ptr)
'maintain tree balance after inserting node x 'check Red-Black properties While (x <> This.Root Andalso x->parent->kolor = RED) 'we have a violation If (x->parent = x->parent->parent->izda) Then Var y = x->parent->parent->dcha If (y->kolor = RED) Then 'uncle is RED x->parent->kolor = BLACK y->kolor = BLACK x->parent->parent->kolor = RED x = x->parent->parent Else 'uncle is BLACK If (x = x->parent->dcha) Then 'make x a izda child x = x->parent This.rotateLeft(x) End If 'recolor and rotate x->parent->kolor = BLACK x->parent->parent->kolor = RED This.rotateRight(x->parent->parent) End If Else ' mirror image of above code Var y = x->parent->parent->izda If (y->kolor = RED) Then ' uncle is RED x->parent->kolor = BLACK y->kolor = BLACK x->parent->parent->kolor = RED x = x->parent->parent Else ' uncle is BLACK If (x = x->parent->izda) Then x = x->parent This.rotateRight(x) End If x->parent->kolor = BLACK x->parent->parent->kolor = RED This.rotateLeft(x->parent->parent) End If End If Wend This.root->kolor = BLACK
End Sub
Function RBTree.insertNode(Byval key As Integer, value As String) As RBNode Ptr
'Insert a node in the RBTree 'find where node belongs Dim current As RBNode Ptr = This.root Dim parent As RBNode Ptr While (current <> This.sentinel) Var rc = This.Compare(key, current->key) If (rc = 0) Then Return current parent = current If (rc < 0) Then current = current->izda Else current = current->dcha End If Wend ' setup new node Dim x As RBNode Ptr = New RBNode(key, value) x->izda = This.sentinel x->dcha = This.sentinel x->parent = parent This.count = This.count + 1 ' insert node in tree If (parent) Then If (This.Compare(key, parent->key) < 0) Then parent->izda = x Else parent->dcha = x End If Else This.root = x End If This.insertFixup(x) Return x
End Function
Sub RBTree.deleteFixup(Byval x As RBNode Ptr)
'maintain tree balance after deleting node x Dim w As RBNode Ptr While (x <> This.root Andalso x->kolor = BLACK) If (x = x->parent->izda) Then w = x->parent->dcha If (w->kolor = RED) Then w->kolor = BLACK x->parent->kolor = RED This.rotateLeft(x->parent) w = x->parent->dcha End If If (w->izda->kolor = BLACK And w->dcha->kolor = BLACK) Then w->kolor = RED x = x->parent Else If (w->dcha->kolor = BLACK) Then w->izda->kolor = BLACK w->kolor = RED This.rotateRight(w) w = x->parent->dcha End If w->kolor = x->parent->kolor x->parent->kolor = BLACK w->dcha->kolor = BLACK This.rotateLeft(x->parent) x = This.root End If Else w = x->parent->izda If (w->kolor = RED) Then w->kolor = BLACK x->parent->kolor = RED This.rotateRight(x->parent) w = x->parent->izda End If If (w->dcha->kolor = BLACK And w->izda->kolor = BLACK) Then w->kolor = RED x = x->parent Else If (w->izda->kolor = BLACK) Then w->dcha->kolor = BLACK w->kolor = RED This.rotateLeft(w) w = x->parent->izda End If w->kolor = x->parent->kolor x->parent->kolor = BLACK w->izda->kolor = BLACK This.rotateRight(x->parent) x = This.root End If End If Wend x->kolor = BLACK
End Sub
Sub RBTree.deleteNode(Byval z As RBNode Ptr)
'delete node z from tree Dim y As RBNode Ptr Dim x As RBNode Ptr If (0 = z Orelse z = This.sentinel) Then Return If (z->izda = This.sentinel Orelse z->dcha = This.sentinel) Then 'y has a This.sentinel node as a child y = z Else 'find tree successor with a This.sentinel node as a child y = z->dcha While (y->izda <> This.sentinel) y = y->izda Wend End If 'x is y's only child If (y->izda <> This.sentinel) Then x = y->izda Else x = y->dcha End If 'remove y from the parent chain x->parent = y->parent If (y->parent) Then If (y = y->parent->izda) Then y->parent->izda = x Else y->parent->dcha = x End If Else This.root = x End If If (y <> z) Then z->key = y->key z->value = y->value End If If (y->kolor = BLACK) Then This.deleteFixup(x) End If Delete y This.count = This.count - 1
End Sub
Function RBtree.findNode(Byval key As Integer) As RBNode Ptr
'find node with key equal to key Var current = This.root While (current <> This.sentinel) Var rc = This.Compare(key, current->key) If (rc = 0) Then Return current Else If (rc < 0) Then current = current->izda Else current = current->dcha End If End If Wend Return 0
End Function
Type GraphicsNode
Dim node As RBNode Ptr Dim lvl As Ubyte Dim nxt As GraphicsNode Ptr Dim prev As GraphicsNode Ptr Dim x As Uinteger Dim y As Uinteger
End Type
Type NodeQueue
Dim startx As Integer Dim starty As Integer Dim first As GraphicsNode Ptr Dim last As GraphicsNode Ptr Dim levels(2 To 11) As Integer => {100,50,25,12,10,10,10,10,10} Dim count As Integer Declare Constructor Declare Destructor Declare Function Enqueue(Byref item As GraphicsNode Ptr) As Integer Declare Function Dequeue(Byref item As GraphicsNode Ptr) As GraphicsNode Ptr Declare Sub PrintNode(Byval item As GraphicsNode Ptr, Byval x As Integer, Byval y As Integer) Declare Sub PrintTree(Byval tree As RBTree Ptr)
End Type
Constructor NodeQueue() Draw first node in the middle of the screen '(just below the top of the screen)
This.startx = 350 This.starty = 100 This.first = NULL This.last = NULL This.count = 1 '800x600, 32 bits kolor Screen 19,32 color , Rgb(255,255,155) Cls WindowTitle "Red black tree sort"
End Constructor
Destructor NodeQueue()
End Destructor
Function NodeQueue.Enqueue(Byref item As GraphicsNode Ptr) As Integer
'Insertion into an empty que If (This.first = NULL) Then This.first = item This.last = item This.Count += 1 Return 0 Else Var tmp = This.last This.last = item This.last->prev = tmp tmp->nxt = This.last This.last->nxt = NULL This.Count += 1 Return 0 End If Return -1
End Function
Function NodeQueue.Dequeue(Byref item As GraphicsNode Ptr) As GraphicsNode Ptr
'Dequeueing from an empty queue or a queue with one node If (This.last = This.first) Then 'Dequeueing from an empty queue If (This.last = NULL) Then This.Count -= 1 Return NULL Else 'Dequeueing from a queue with one node item->node = This.First->node item->x = This.First->x item->y = This.First->y item->lvl = This.first->lvl Delete This.first This.first = NULL This.last = NULL This.Count -= 1 Return item End If Else 'Dequeueing from a queue with more than one node Var tmp = This.Last item->node = This.Last->node item->x = This.Last->x item->y = This.Last->y item->lvl = This.Last->lvl This.last = This.last->prev This.last->nxt = NULL Delete tmp Return item End If Return NULL
End Function
Sub NodeQueue.PrintNode(Byval item As GraphicsNode Ptr, Byval x As Integer, Byval y As Integer)
'Draw a black line from parent node to child node Line (x,y)-(item->x,item->y), Rgb(0,0,0) 'Draw node (either red or black) If (item->node->kolor = RED) Then Circle (item->x,item->y),5, Rgb(255,0,0),,,,F Else Circle (item->x,item->y),5, Rgb(0,0,0),,,,F End If Draw String (item->x,item->y - 40), Str(item->node->key), Rgb(0,0,0) Draw String (item->x-8,item->y - 25),"""" & item->node->value & """", Rgb(0,0,0)
End Sub
Sub NodeQueue.PrintTree(Byval tree As RBTree Ptr)
Dim item As GraphicsNode Ptr Dim current As GraphicsNode Ptr = New GraphicsNode Dim tmp As GraphicsNode Ptr Dim lvl As Integer = 1 Dim x As Integer = This.startx Dim y As Integer = This.starty 'check for empty tree If (tree->root = tree->sentinel) Then Return 'Start with printing the root current->node = tree->root current->x = x current->y = y current->lvl = lvl This.PrintNode(current,x,y) Do 'Print izda node (position it at izda side of current node) If (current->node->izda <> tree->sentinel) Then item = New GraphicsNode item->lvl = lvl + 1 If (item->lvl <= 9) Then item->x = x - This.levels(lvl+1) Else item->x = x - 10 End If item->y = y + 50 item->node = current->node->izda This.PrintNode(item,x,y) This.Enqueue(item) End If 'Print dcha node (position it at dcha side of current node If (current->node->dcha <> tree->sentinel) Then item = New GraphicsNode item->lvl = lvl + 1 If (item->lvl <= 9) Then item->x = x + This.levels(lvl+1) Else item->x = x + 10 End If item->y = y + 50 item->node = current->node->dcha This.PrintNode(item,x,y) This.Enqueue(item) End If 'Continue drawing from first node in the queue 'Nodes in izda tree will be drawn first as these are put in 'the queue first Var tmp = This.Dequeue(current) 'If count smaller then entire tree has been drawn If (This.count < 1) Then Exit Do x = current->x y = current->y lvl = current->lvl Loop
End Sub
Dim x As Integer Ptr Dim i As Integer Var tree = New RBTree(@integer_compare)
Open Cons For Output As #1 For i = 0 To 29
Print #1, "Insert "; i tree->Insertnode(i,Str(i)) Sleep() Dim print_tree As NodeQueue Ptr print_tree = New NodeQueue print_tree->PrintTree(tree) Delete print_tree
Next i
Print #1, !"\nStarting Deletion after keypress" Var print_tree = New NodeQueue print_tree->PrintTree(tree) Sleep() Delete print_tree
randomize timer For i = 0 To 14
Dim as integer j = int(rnd * 15) + int(rnd * 16) Print #1, "Delete"; j Var n = tree->FindNode(j) If (n) Then tree->Deletenode(n) Sleep() Dim print_tree As NodeQueue Ptr print_tree = New NodeQueue print_tree->PrintTree(tree) Delete print_tree
Next i
Bsave "FreeBASIC_Red-black-tree_sort.bmp", 0 Print #1, !"\nEnding program after keypress" Sleep() Close #1 Delete tree</lang>
- Output:
FreeBasic Red black tree sort image
Julia
See Red_black_tree_sort/Julia.
Phix
Python
This is based on the code of Shivali Bhadaniya [1], which I thought was reasonably intelligible. <lang python>#!/usr/bin/python
- Define Node
class Node():
def __init__(self,val): self.val = val # Value of Node self.parent = None # Parent of Node self.left = None # Left Child of Node self.right = None # Right Child of Node self.color = 1 # Red Node as new node is always inserted as Red Node
- Define R-B Tree
class RBTree():
def __init__(self): self.NULL = Node ( 0 ) self.NULL.color = 0 self.NULL.left = None self.NULL.right = None self.root = self.NULL
# Insert New Node def insertNode(self, key): node = Node(key) node.parent = None node.val = key node.left = self.NULL node.right = self.NULL node.color = 1 # Set root colour as Red
y = None x = self.root
while x != self.NULL : # Find position for new node y = x if node.val < x.val : x = x.left else : x = x.right
node.parent = y # Set parent of Node as y if y == None : # If parent i.e, is none then it is root node self.root = node elif node.val < y.val : # Check if it is right Node or Left Node by checking the value y.left = node else : y.right = node
if node.parent == None : # Root node is always Black node.color = 0 return
if node.parent.parent == None : # If parent of node is Root Node return
self.fixInsert ( node ) # Else call for Fix Up
def minimum(self, node): while node.left != self.NULL: node = node.left return node
# Code for left rotate def LR ( self , x ) : y = x.right # Y = Right child of x x.right = y.left # Change right child of x to left child of y if y.left != self.NULL : y.left.parent = x
y.parent = x.parent # Change parent of y as parent of x if x.parent == None : # If parent of x == None ie. root node self.root = y # Set y as root elif x == x.parent.left : x.parent.left = y else : x.parent.right = y y.left = x x.parent = y
# Code for right rotate def RR ( self , x ) : y = x.left # Y = Left child of x x.left = y.right # Change left child of x to right child of y if y.right != self.NULL : y.right.parent = x
y.parent = x.parent # Change parent of y as parent of x if x.parent == None : # If x is root node self.root = y # Set y as root elif x == x.parent.right : x.parent.right = y else : x.parent.left = y y.right = x x.parent = y
# Fix Up Insertion def fixInsert(self, k): while k.parent.color == 1: # While parent is red if k.parent == k.parent.parent.right: # if parent is right child of its parent u = k.parent.parent.left # Left child of grandparent if u.color == 1: # if color of left child of grandparent i.e, uncle node is red u.color = 0 # Set both children of grandparent node as black k.parent.color = 0 k.parent.parent.color = 1 # Set grandparent node as Red k = k.parent.parent # Repeat the algo with Parent node to check conflicts else: if k == k.parent.left: # If k is left child of it's parent k = k.parent self.RR(k) # Call for right rotation k.parent.color = 0 k.parent.parent.color = 1 self.LR(k.parent.parent) else: # if parent is left child of its parent u = k.parent.parent.right # Right child of grandparent if u.color == 1: # if color of right child of grandparent i.e, uncle node is red u.color = 0 # Set color of childs as black k.parent.color = 0 k.parent.parent.color = 1 # set color of grandparent as Red k = k.parent.parent # Repeat algo on grandparent to remove conflicts else: if k == k.parent.right: # if k is right child of its parent k = k.parent self.LR(k) # Call left rotate on parent of k k.parent.color = 0 k.parent.parent.color = 1 self.RR(k.parent.parent) # Call right rotate on grandparent if k == self.root: # If k reaches root then break break self.root.color = 0 # Set color of root as black
# Function to fix issues after deletion def fixDelete ( self , x ) : while x != self.root and x.color == 0 : # Repeat until x reaches nodes and color of x is black if x == x.parent.left : # If x is left child of its parent s = x.parent.right # Sibling of x if s.color == 1 : # if sibling is red s.color = 0 # Set its color to black x.parent.color = 1 # Make its parent red self.LR ( x.parent ) # Call for left rotate on parent of x s = x.parent.right # If both the child are black if s.left.color == 0 and s.right.color == 0 : s.color = 1 # Set color of s as red x = x.parent else : if s.right.color == 0 : # If right child of s is black s.left.color = 0 # set left child of s as black s.color = 1 # set color of s as red self.RR ( s ) # call right rotation on x s = x.parent.right
s.color = x.parent.color x.parent.color = 0 # Set parent of x as black s.right.color = 0 self.LR ( x.parent ) # call left rotation on parent of x x = self.root else : # If x is right child of its parent s = x.parent.left # Sibling of x if s.color == 1 : # if sibling is red s.color = 0 # Set its color to black x.parent.color = 1 # Make its parent red self.RR ( x.parent ) # Call for right rotate on parent of x s = x.parent.left
if s.right.color == 0 and s.right.color == 0 : s.color = 1 x = x.parent else : if s.left.color == 0 : # If left child of s is black s.right.color = 0 # set right child of s as black s.color = 1 self.LR ( s ) # call left rotation on x s = x.parent.left
s.color = x.parent.color x.parent.color = 0 s.left.color = 0 self.RR ( x.parent ) x = self.root x.color = 0
# Function to transplant nodes def __rb_transplant ( self , u , v ) : if u.parent == None : self.root = v elif u == u.parent.left : u.parent.left = v else : u.parent.right = v v.parent = u.parent
# Function to handle deletion def delete_node_helper ( self , node , key ) : z = self.NULL while node != self.NULL : # Search for the node having that value/ key and store it in 'z' if node.val == key : z = node
if node.val <= key : node = node.right else : node = node.left
if z == self.NULL : # If Kwy is not present then deletion not possible so return print ( "Value not present in Tree !!" ) return
y = z y_original_color = y.color # Store the color of z- node if z.left == self.NULL : # If left child of z is NULL x = z.right # Assign right child of z to x self.__rb_transplant ( z , z.right ) # Transplant Node to be deleted with x elif (z.right == self.NULL) : # If right child of z is NULL x = z.left # Assign left child of z to x self.__rb_transplant ( z , z.left ) # Transplant Node to be deleted with x else : # If z has both the child nodes y = self.minimum ( z.right ) # Find minimum of the right sub tree y_original_color = y.color # Store color of y x = y.right if y.parent == z : # If y is child of z x.parent = y # Set parent of x as y else : self.__rb_transplant ( y , y.right ) y.right = z.right y.right.parent = y
self.__rb_transplant ( z , y ) y.left = z.left y.left.parent = y y.color = z.color if y_original_color == 0 : # If color is black then fixing is needed self.fixDelete ( x )
# Deletion of node def delete_node ( self , val ) : self.delete_node_helper ( self.root , val ) # Call for deletion
# Function to print def __printCall ( self , node , indent , last ) : if node != self.NULL : print(indent, end=' ') if last : print ("R----",end= ' ') indent += " " else : print("L----",end=' ') indent += "| "
s_color = "RED" if node.color == 1 else "BLACK" print ( str ( node.val ) + "(" + s_color + ")" ) self.__printCall ( node.left , indent , False ) self.__printCall ( node.right , indent , True )
# Function to call print def print_tree ( self ) : self.__printCall ( self.root , "" , True )
if __name__ == "__main__":
bst = RBTree()
print("State of the tree after inserting the 30 keys:") for x in range(1, 30): bst.insertNode(x) bst.print_tree()
print("\nState of the tree after deleting the 15 keys:") for x in range(1, 15): bst.delete_node(x) bst.print_tree()
</lang>
- Output:
State of the tree after inserting the 30 keys: R---- 8(BLACK) L---- 4(BLACK) | L---- 2(BLACK) | | L---- 1(BLACK) | | R---- 3(BLACK) | R---- 6(BLACK) | L---- 5(BLACK) | R---- 7(BLACK) R---- 16(BLACK) L---- 12(RED) | L---- 10(BLACK) | | L---- 9(BLACK) | | R---- 11(BLACK) | R---- 14(BLACK) | L---- 13(BLACK) | R---- 15(BLACK) R---- 20(RED) L---- 18(BLACK) | L---- 17(BLACK) | R---- 19(BLACK) R---- 24(BLACK) L---- 22(RED) | L---- 21(BLACK) | R---- 23(BLACK) R---- 26(RED) L---- 25(BLACK) R---- 28(BLACK) L---- 27(RED) R---- 29(RED) State of the tree after deleting the 15 keys: R---- 20(BLACK) L---- 16(BLACK) | L---- 15(BLACK) | R---- 18(RED) | L---- 17(BLACK) | R---- 19(BLACK) R---- 24(BLACK) L---- 22(RED) | L---- 21(BLACK) | R---- 23(BLACK) R---- 26(RED) L---- 25(BLACK) R---- 28(BLACK) L---- 27(RED) R---- 29(RED)