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Dr. S. Lovelyn Rose

PSG College of Technology

Coimbatore, India

Binary Search TreesInsertion and Deletion

(using Arrays and Linked lists)

Algorithm Insert_LinkedList_BST(root, element)

* root points to the root element of the linked list* ‘element’ is the element to the inserted* ‘data’ holds the data part of a node* lchild stores the address of the left child* rchild stores the address of the right child* GetNode() is a function that creates a node with a

data part and 2 address parts and assigns null in the address parts

Insertion in a Binary Search Tree Using Linked Lists

1.temp=root; 2 if(root == null)3. rootdata = element;4. Return;5. end if6. while(1) 7. if(tempdata >

element)8. if(templchild ==

null)9.templchild =

GetNode()10.templchild data

= element10.break;11.end if

Insert as first node

2

1

0

5

83

Element to be inserted is 4

temp

12.else13.temp = templchild14.end else15.end if16.else17.if(temprchild ==

null)18.temprchild =

GetNode()19.temprchild data

= element20.break;21.end if

2

1

0

5

83

Element to be inserted is 4

temp

12.else13.temp = temprchild //Move to the right child23.end else24.end if25.end while26. end Insert_LinkedList_BST

2

1

0

5

83

Element to be inserted is 4

temp

Algorithm Delete_LinkedList_BST(root)* root is a pointer to the root node* ‘data’ holds the data part of a node* lchild stores the address of the left child* rchild stores the address of the right child* temp is used to find the address of the

element to be deleted* parent stores the address of the parent of the

element to be deleted* inorder_N is the inorder successor* parent_inorder_N is the parent of the inorder

successor

Delete a node in a BST (using Linked List)

//Search for the element

1. temp = root

2. parent = root

3. while((element != tempdata) && (temp != null))

// Loop until the element is found

4.parent = temp

5.if(element < tempdata)

6.temp = temp left // Move left

7.end if

8.else

9.temp = tempright //Move right

10.end else

11.end while

5

3

2 4

7

9

Element to be searched

is 4

12. If(temp != null)

13.if((templchild == null) and (temprchild == null))

14.if(parent == temp)

15.root = null

16.end if

17. else if(temp == parentlchild)

18. parentlchild = null

19. end else if

else

21.parentrchild = null

22.end else

23.FreeNode(temp)

24.end if

Element found

temp is a leaf node

Only one node

5

4 73

2 4

// Case 2 - Delete node with one child

else if((templchild == null)or(temprchild == null))

26.if(parent == temp)

27.if(templchild != null)

28.root = templchild

29.end if

30.else

31.root = temprchild

32.end else

33.end if

4

2

root

else if(temp == parentlchild)

35.if(templchild != null)

36.parentlchild = templchild

37.end if

38.else

39.parentlchild = temprchild

40.end else

41.end else if

else

43.if(templchild != null)

44.parentrchild = templchild

45.end if

46.else

47. parentrchild = temprchild

48.end else

49.end else if

50.FreeNode(temp)

51.end else if

5

4

2

7

9

temp is a left child

temp is a right child

//Case 3 – Delete node with 2 children

52.else53.parent_inorder_N = temp54.inorder_N = tempright subtree55.Repeat steps 56,57

while inorder_Nleft ≠ null56.parent_inorder_N = inorder_N57.inorder_N = inorder_N left58.end Repeat59.tempdata = inorder_Ndata60. if(inorder_N rchild == null)61. parent_ inorder_Nleft = null62. end if

2

1

0

4

53

6

Parent_inorder_N

inorder_N

6

63.else // Inorder successor has one child64. parent_ inorder_Nleft = inorder_Nright65.end else66. end else67.end if68.else69.Print “Element not Found”70.end else71.end Delete_LinkedList_BST

Algorithm Insert_Array_BST(Tree[1:N], element)

* Tree is an array of n elements representing the BST

* ‘element’ is the element to the inserted* The empty positions in an array are denoted by -1

Insertion in a Binary Search Tree Using Arrays

1. temp=1;2. if(Tree[temp] == -1)3.Tree[temp] = element;4. Return;5. end if6. while(1)7. if(Tree[temp] > element)8. if(Tree[2 * temp] == -1)9. Tree[2 * temp] ==

element10.break;11. end if

10 15 -1 -1 -1 -14

Element to be inserted is 6

Insert as first node

To find the position

12.else13.temp = 2 * temp14.end else15.end if16. else17. if(Tree[2 * temp+1]

== -1)18.Tree[2 * temp+1]

== element child19.break;20.end if

6 -1-1

21. else // The right child is occupied22.temp = 2 * temp + 1 //Move to the right child23. end else24. end if25. end while26. end Insert_Array_BST

Algorithm Delete_Array_BST(Tree[1:N], element)

* Tree is an array of n elements representing the BST

* ‘element’ is the element to the inserted

* The empty positions in an array are denoted by -1

* temp is used to find the subscript of the element to be deleted

* inorder_N is the inorder successor

Delete a node in a BST using Arrays

//Search for the ‘element’ in the BST

1. temp = 12. while((Tree[temp]!=element) && (Tree[temp] !=

-1)) // Loop until the element is found3.if(element < Tree[temp])4.temp = 2 * temp // Move left5.end if6. else7.temp = 2 * temp + 1 //Move right8. end else9. end while 10 4 15 -1 -1 -1 -1

Element to be searched is 4

temp

10. if(Tree[temp] != -1) // If the element is found

// Case 1 - Delete leaf node

11.if((Tree[2*temp] == -1) and (Tree[2*temp+1] == -1))

12.Tree[temp] = -113.end if

10 4 15 -1 -1 -1 -1

temp

-1

// Case 2 - Delete node with one child

14.else if((Tree[2*temp] == -1) or (Tree[2*temp+1] == -1))

15.if(Tree[2*temp] != -1) // Is the child in the left of temp

16.Call Preorder(Tree[1:N], 2*temp) //Update the whole subtree

17.end if18.else19.Call Preorder(Tree[1:N], 2*temp+1)20.end else21.end else if

10 15 -1 7 -1 -1

temp

4 -1-1

//Case 3 – Delete node with 2 children22.else23.inorder_N = 2*temp+1 // Inorder successor is

surely in the right subtree24.Repeat steps 48,49 while Tree[2*inorder_N] ≠ -125.inorder_N = 2*inorder_N26.end Repeat27.Tree[temp] = Tree[inorder_N] // Replace with

inorder successor28.if(Tree[2*inorder_N + 1] == -1) // Inorder

successor has no child29.Tree[inorder_N] = -130.end if

31.else // Inorder successor has one child32.Call Preorder(Tree[1:N], 2*inorder_N+1)33.end else34.end else35.end if36.else37.Print “Element not Found”38.end else39.end Delete_Array_BST

Algorithm Insert_LinkedList_BST(root, element)

* root points to the root element of the linked list* ‘element’ is the element to the inserted* ‘data’ holds the data part of a node* lchild stores the address of the left child* rchild stores the address of the right child* GetNode() is a function that creates a node with a

data part and 2 address parts and assigns null in the address parts

Insertion in a Binary Search Tree Using Linked Lists

1.temp=root; 2 if(root == null)3. rootdata = element;4. Return;5. end if6. while(1) 7. if(tempdata >

element)8. if(templchild ==

null)9.templchild =

GetNode()10.templchild data

= element10.break;11.end if

Insert as first node

2

1

0

5

83

Element to be inserted is 4

temp

12.else13.temp = templchild14.end else15.end if16.else17.if(temprchild ==

null)18.temprchild =

GetNode()19.temprchild data

= element20.break;21.end if

2

1

0

5

83

Element to be inserted is 4

temp

12.else13.temp = temprchild //Move to the right child23.end else24.end if25.end while26. end Insert_LinkedList_BST

2

1

0

5

83

Element to be inserted is 4

temp

Algorithm Delete_LinkedList_BST(root)* root is a pointer to the root node* ‘data’ holds the data part of a node* lchild stores the address of the left child* rchild stores the address of the right child* temp is used to find the address of the

element to be deleted* parent stores the address of the parent of the

element to be deleted* inorder_N is the inorder successor* parent_inorder_N is the parent of the inorder

successor

Delete a node in a BST (using Linked List)

//Search for the element

1. temp = root

2. parent = root

3. while((element != tempdata) && (temp != null))

// Loop until the element is found

4.parent = temp

5.if(element < tempdata)

6.temp = temp left // Move left

7.end if

8.else

9.temp = tempright //Move right

10.end else

11.end while

5

3

2 4

7

9

Element to be searched

is 4

12. If(temp != null)

13.if((templchild == null) and (temprchild == null))

14.if(parent == temp)

15.root = null

16.end if

17. else if(temp == parentlchild)

18. parentlchild = null

19. end else if

else

21.parentrchild = null

22.end else

23.FreeNode(temp)

24.end if

Element found

temp is a leaf node

Only one node

5

4 73

2 4

// Case 2 - Delete node with one child

else if((templchild == null)or(temprchild == null))

26.if(parent == temp)

27.if(templchild != null)

28.root = templchild

29.end if

30.else

31.root = temprchild

32.end else

33.end if

4

2

root

else if(temp == parentlchild)

35.if(templchild != null)

36.parentlchild = templchild

37.end if

38.else

39.parentlchild = temprchild

40.end else

41.end else if

else

43.if(templchild != null)

44.parentrchild = templchild

45.end if

46.else

47. parentrchild = temprchild

48.end else

49.end else if

50.FreeNode(temp)

51.end else if

5

4

2

7

9

temp is a left child

temp is a right child

//Case 3 – Delete node with 2 children

52.else53.parent_inorder_N = temp54.inorder_N = tempright subtree55.Repeat steps 56,57

while inorder_Nleft ≠ null56.parent_inorder_N = inorder_N57.inorder_N = inorder_N left58.end Repeat59.tempdata = inorder_Ndata60. if(inorder_N rchild == null)61. parent_ inorder_Nleft = null62. end if

2

1

0

4

53

6

Parent_inorder_N

inorder_N

6

63.else // Inorder successor has one child64. parent_ inorder_Nleft = inorder_Nright65.end else66. end else67.end if68.else69.Print “Element not Found”70.end else71.end Delete_LinkedList_BST

Algorithm Insert_Array_BST(Tree[1:N], element)

* Tree is an array of n elements representing the BST

* ‘element’ is the element to the inserted* The empty positions in an array are denoted by -1

Insertion in a Binary Search Tree Using Arrays

1. temp=1;2. if(Tree[temp] == -1)3.Tree[temp] = element;4. Return;5. end if6. while(1)7. if(Tree[temp] > element)8. if(Tree[2 * temp] == -1)9. Tree[2 * temp] ==

element10.break;11. end if

10 15 -1 -1 -1 -14

Element to be inserted is 6

Insert as first node

To find the position

12.else13.temp = 2 * temp14.end else15.end if16. else17. if(Tree[2 * temp+1]

== -1)18.Tree[2 * temp+1]

== element child19.break;20.end if

6 -1-1

21. else // The right child is occupied22.temp = 2 * temp + 1 //Move to the right child23. end else24. end if25. end while26. end Insert_Array_BST

Algorithm Delete_Array_BST(Tree[1:N], element)

* Tree is an array of n elements representing the BST

* ‘element’ is the element to the inserted

* The empty positions in an array are denoted by -1

* temp is used to find the subscript of the element to be deleted

* inorder_N is the inorder successor

Delete a node in a BST using Arrays

//Search for the ‘element’ in the BST

1. temp = 12. while((Tree[temp]!=element) && (Tree[temp] !=

-1)) // Loop until the element is found3.if(element < Tree[temp])4.temp = 2 * temp // Move left5.end if6. else7.temp = 2 * temp + 1 //Move right8. end else9. end while 10 4 15 -1 -1 -1 -1

Element to be searched is 4

temp

10. if(Tree[temp] != -1) // If the element is found

// Case 1 - Delete leaf node

11.if((Tree[2*temp] == -1) and (Tree[2*temp+1] == -1))

12.Tree[temp] = -113.end if

10 4 15 -1 -1 -1 -1

temp

-1

// Case 2 - Delete node with one child

14.else if((Tree[2*temp] == -1) or (Tree[2*temp+1] == -1))

15.if(Tree[2*temp] != -1) // Is the child in the left of temp

16.Call Preorder(Tree[1:N], 2*temp) //Update the whole subtree

17.end if18.else19.Call Preorder(Tree[1:N], 2*temp+1)20.end else21.end else if

10 15 -1 7 -1 -1

temp

4 -1-1

//Case 3 – Delete node with 2 children22.else23.inorder_N = 2*temp+1 // Inorder successor is

surely in the right subtree24.Repeat steps 48,49 while Tree[2*inorder_N] ≠ -125.inorder_N = 2*inorder_N26.end Repeat27.Tree[temp] = Tree[inorder_N] // Replace with

inorder successor28.if(Tree[2*inorder_N + 1] == -1) // Inorder

successor has no child29.Tree[inorder_N] = -130.end if

31.else // Inorder successor has one child32.Call Preorder(Tree[1:N], 2*inorder_N+1)33.end else34.end else35.end if36.else37.Print “Element not Found”38.end else39.end Delete_Array_BST

Algorithm Preorder(Tree[1:N], root)* Tree is the array holding the tree* root is the subscript of the root node* The empty positions in an array are denoted by -1

1. Tree[ceil(root/2)-1] = Tree[root]2. Tree[root] = -13. if(Tree[2*root] ≠ -1) // A left child exists4. call Preorder(Tree[], 2 * root)5. end if6. if ( Tree[2*root+1] ≠ -1) // Does a right child

exist7. call Preorder(Tree[], 2 * root+1)8. end if9. end Preorder

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