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Objectives

In this session, you will learn to:

Implement a threaded binary tree

Implement a height balanced binary tree

Store data in a graph

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Deleting a Node from a Threaded Binary Tree

Delete operation in a threaded binary tree refers to theprocess of removing the specified node from the threadedbinary tree.

Before implementing a delete operation, you first need to

locate the node to be deleted.This requires you to implement a search operation.

After the search operation, the node to be deleted is markedas the currentNode and its parent as parent.

Write an algorithm to locate the node to be deleted in athreaded binary tree.

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header node

. .

.

Algorithm to locate the nodeto be deleted and its parent ina threaded binary tree.

1. Mark the left child of the header node ascurrentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

Delete 80

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

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Delete 80

currentNode

1. Mark the left child of the header node ascurrentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

. .

.

Delete 80

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

currentNode

parent

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

currentNode parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

currentNode parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

. .

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Delete 80

currentNode parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the node

to be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

currentNode parent

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

parent

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

parent

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

. .

.

Delete 80

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

. .

.

Delete 80

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

. .

.

Delete 80

currentNode

parent

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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.

Delete 80

currentNode

parent

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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header node

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Delete 80

parent

currentNode

Nodes located

Deleting a Node from a Threaded Binary Tree (Contd.)1. Mark the left child of the header node as

currentNode.

2. Mark the header node as parent.3. Repeat steps a, b, c, d, and e until the nodeto be searched is found or currentNodebecomes NULL:a. Mark currentNode as parent.b. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a link:i. Make currentNode point to its

left child and go to step 3.c. If the value to be searched is less

than that of currentNode, and the leftchild of the currentNode is a thread:i. Make currentNode as NULL

and go to step 3.d. If the value to be searched is greater

than that of currentNode, and theright child of the currentNode is a link:

i. Make currentNode point to itsright child and go to step 3.

e. If the value to be searched is greaterthan that of currentNode, and theright child of the currentNode is athread:i. Mark currentNode as NULL

and go to step 3.

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Once you locate the node to be deleted and its parent, youcan release the memory of the node to be deleted afteradjusting the links and threads appropriately.

Before implementing a delete operation, you first need tocheck whether the tree is empty or not.

The tree is empty if the left child of the header node is athread pointing to itself.

If the tree is empty, an error message is shown.

However, if the tree is not empty, there can be three cases:

Node to be deleted is a leaf node

Node to be deleted has one child (left or right)

Node to be deleted has two children

header node

Deleting a Node from a Threaded Binary Tree (Contd.)

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Let us first consider a case in which the node to be deletedis the leaf node.

In this case, you first need to check if there is only one nodepresent in the tree.

header node

65

new node

Deleting a Node from a Threaded Binary Tree (Contd.)

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To delete this node, make the left child of the header nodeas a thread pointing to itself.

header node

65

new node

Deleting a Node from a Threaded Binary Tree (Contd.)

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To delete this node, make the left child of the header nodeas a thread pointing to itself.

Now release the memory of the node to be deleted.

header node

65

new node

Delete operation complete

Deleting a Node from a Threaded Binary Tree (Contd.)

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However, if there are more thanone nodes in the tree, you needanother algorithm to delete a leafnode.

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

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Delete node 60

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

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.

Delete node 60

currentNode

parent

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

. .

.

Delete node 60

currentNode

parent

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

. .

.

Delete node 60

currentNode

parent

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

. .

.

Delete node 60

currentNode

parent

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

. .

.

Delete node 60

currentNode

parent

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

.

.

Delete node 60

currentNode

parent

Deletion complete

1. Locate the position of the node to

be deleted. Mark it as currentNodeand its parent as parent.

2. If currentNode is the left child ofparent:a. Set the left thread field of

parent as zero.b. Make the left child field of

parent as a thread pointing

to the inorder predecessorof currentNode.

c. Go to step 4.

3. If currentNode is the right child ofparent:a. Set the right thread field of

parent as zero.b. Make the right child field of

parent as a thread pointingto the inorder successor ofcurrentNode.

4. Release the memory forcurrentNode.

Deleting a Node from a Threaded Binary Tree (Contd.)

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Session 16Ver. 1.0

Write an algorithm to delete a node, which has one child in athreaded binary tree.

Deleting a Node from a Threaded Binary Tree (Contd.)

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Session 16Ver. 1.0

Let us now consider a case inwhich the node to be deletedhas one child (left or right).

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header node

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.

Algorithm to delete a nodewith one child from athreaded binary tree.

Delete node 50

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successorpoint to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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.parent

currentNode

Delete node 50

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

S d l i h

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header node

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.

Delete node 50

parent

currentNode

child

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

D S d Al i h

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header node

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.

Delete node 50

parent

currentNode

child

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

D t St t d Al ith

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header node

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.

Delete node 50

parent

currentNode

child

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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.

Delete node 50

parent

currentNode

child

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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.

Delete node 50

parent

currentNode

child

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

Data St t e a d Al o ith

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Session 16Ver. 1.0

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header node

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Delete node 50

parent

currentNode

childsuccessor

predecessor

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

currentNode

childsuccessor

predecessor

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

currentNode

childsuccessor

predecessor

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

childsuccessor

predecessor

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node aschild.

b. Go to step 4.3. If currentNode has a right subtree:

a. Mark the right child of currentNode aschild.

b. Go to step 4.4. If currentNode is the left child of parent:

a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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header node

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Delete node 50

parent

childsuccessor

predecessor

currentNodeDeletion

complete

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the node to be deleted. Mark it ascurrentNode and its parent as parent.

2. If currentNode has a left subtree:a. Mark the left child of current node as

child.b. Go to step 4.

3. If currentNode has a right subtree:a. Mark the right child of currentNode as

child.b. Go to step 4.

4. If currentNode is the left child of parent:a. Make the left child of parent point to

child.b. Go to step 6.

5. If currentNode is the right child of parent:a. Make the right child of parent point to

child.b. Go to step 6.

6. Find the inorder successor and inorderpredecessor of currentNode. Mark them assuccessor and predecessor, respectively.

7. If currentNode has a right child:a. Make the left child field of its successor

point to its predecessor and set the leftthread of successor to zero.

b. Go to step 9.8. If currentNode has a left child:

a. Make the right child of its predecessorpoint to its successor and set the rightthread of predecessor to zero.

9. Release the memory for currentNode.

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Data Structures and Algorithms

Session 16Ver. 1.0

Write an algorithm to delete a node, which has two childrenfrom a threaded binary tree.Algorithm to delete a node havingtwo children from a threaded binarytree.

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

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Delete node 40 1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

Deleting a Node from a Threaded Binary Tree (Contd.)

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header node

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.

Delete node 40

currentNode

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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header node

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.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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header node

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.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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header node

.

.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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header node

.

Delete node 40

currentNode

Inorder_suc

Inorder_parent

Deletion complete

Deleting a Node from a Threaded Binary Tree (Contd.)

1. Locate the position of the node to

be deleted. Mark it ascurrentNode.

2. Locate the inorder successor ofcurrentNode. Mark it asInorder_suc and its parent asInorder_parent.

3. Overwrite the data contained in

currentNode with the datacontained in Inorder_suc.

4. Delete the node markedInorder_suc. This node wouldhave at most one child and cantherefore be deleted by using thealgorithm for one of the following:

a. Deleting a leaf node.

b. Deleting a node with onechild.

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Session 16Ver. 1.0

Implementing a Height Balanced Tree

In a binary search tree, the time required to search for aparticular value depends upon its height.

The shorter the height, the faster is the search.

However, binary search trees tend to attain large heightsbecause of continuous insert and delete operations.

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Session 16Ver. 1.0

After inserting values in the specified order, the binarysearch tree appears as follows:Consider an example in which you want to insert somenumeric values in a binary search tree in the following order:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Defining Height Balanced Tree

12

34

56

78

910

1112

1314

15

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As a result, the binary search tree attains a linear structure.

In this case, the height of the binary search tree is 15.

12

34

56

78

910

1112

1314

15

Now if you want to search for a value 14 in the given binarysearch tree, you will have to traverse all its preceding nodesbefore you reach node 14. In this case, you need to make14 comparisons.

In such a case, the binary search tree becomes equivalentto a linked list.

Defining Height Balanced Tree (Contd.)

This process can be very time consuming if the number of valuesstored in a binary search tree is large.

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Therefore, such a structure loses its property of a binarysearch tree in which after every comparison, the searchoperations are reduced to half.

12

34

56

78

910

1112

1314

15

Defining Height Balanced Tree (Contd.)

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To solve this problem, it is desirable to keep the height ofthe tree to a minimum.

Therefore, the following binary search tree can be modifiedto reduce its height.

12

34

56

78

910

1112

1314

15

Defining Height Balanced Tree (Contd.)

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The height of the binary search tree has now been reducedto 4.Now if you want to search for a value 14, you just need totraverse nodes 8 and 12, before you reach node 14.In this case, the total number of comparisons to be made tosearch for node 14 is three.

This approach reduces the time to search for a particularvalue in a binary search tree.

Defining Height Balanced Tree (Contd.)

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ata St uctu es a d go t s

Session 16Ver. 1.0

This can be implemented with the help of a height balancedtree.

Defining Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

A height balanced tree is a binary tree in which thedifference between the heights of the left subtree and rightsubtree of a node is not more than one.

In a height balanced tree, each node has a Balance Factor(BF) associated with it.

For the tree to be balanced, BF can have three values:

0: A BF value of 0 indicates that the height of the left subtree ofa node is equal to the height of its right subtree.

1: A BF value of 1 indicates that the height of the left subtree isgreater than the height of the right subtree by one. A node in

this state is said to be left heavy.1: A BF value of 1 indicates that the height of the rightsubtree is greater than the height of the left subtree by one. Anode in this state is said to be right heavy.

Defining Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

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0

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-1

-1

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0

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1

30

0

1

010

Balanced Binary Search Trees

Defining Height Balanced Tree (Contd.)

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g

Session 16Ver. 1.0

After inserting a new node in a height balanced tree, thebalance factor of one or more nodes may attain a valueother than 1, 0, or 1.

This makes the tree unbalanced.

In such a case, you first need to locate the pivot node.

A pivot node is the nearest ancestor of the newly insertednode, which has a balance factor other than 1, 0 or 1.

To restore the balance, you need to perform appropriaterotations around the pivot node.

Defining Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

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-2

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35

1

34

0

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0

Unbalanced Binary Search Trees

Defining Height Balanced Tree (Contd.)

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g

Session 16Ver. 1.0

In a height balanced tree, the maximum difference betweenthe height of left and right subtree of a node canbe _________.

Just a minute

Answer:

One

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g

Session 16Ver. 1.0

Inserting Nodes in a Height Balanced Tree

Insert operation in a height balanced tree is similar to that ina simple binary search tree.

However, inserting a node can make the tree unbalanced.

To restore the balance, you need to perform appropriaterotations around the pivot node.

This involves two cases:

When the pivot node is initially right heavy and the new node isinserted in the right subtree of the pivot node.

When the pivot node is initiallyleft heavy and the new node isinserted in the left subtree of the pivot node.

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g

Session 16Ver. 1.0

Let us first consider a case in which the pivot node is initiallyright heavy and the new node is inserted in the right subtreeof the pivot node.

In this case, after the insertion of a new node, the balancefactor of pivot node becomes 2.

Now there can be two situations in this case:

If a new node is inserted in the right subtree of the right child ofthe pivot node.

If the new node is inserted in the left subtree of the right childof the pivot node.

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Session 16Ver. 1.0

After insertion of a node, the binary tree appears as:

Inserting Nodes in a Height Balanced Tree (Contd.)

The tree now becomes unbalanced.

P

X

0

Pl

-1

Xl Xr

-1

-2

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Session 16Ver. 1.0

To restore the balance, you need to perform a single leftrotation around the pivot node.

Xr

P

X

Xl

Pl

Xr

0

-1

Xr

X

P

XlPl Xr

After rotation

The tree is now balanced.

Inserting Nodes in a Height Balanced Tree (Contd.)

-1

-2

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Session 16Ver. 1.0

In the second case, a new node is inserted in the leftsubtree of X.

In this case, the binary tree initially appears as:

P

-1

Yl Yr

Pl

X

0

0

Y

Xr

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Session 16Ver. 1.0

When a new node is inserted in the left subtree of X, thebinary tree becomes unbalanced.

P

-1

Yl Yr

Pl

X0

Y

Xr

Yl

-2

-10

1

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Session 16Ver. 1.0

To restore the balance, you need to perform a doublerotation.In the first rotation, the subtree with root X is rotated right insuch a way so that the right child of Y now points to X.

Inserting Nodes in a Height Balanced Tree (Contd.)

P

Yr

Pl

X

Y

Xr

-1

Yl

1

-2

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Session 16Ver. 1.0

P

Yr

Pl

X

Y

Xr

-1

Yl

1

-2

P

-2

Yl

Yr

Pl

X-1

-1

Y

Xr

After single right rotation

Inserting Nodes in a Height Balanced Tree (Contd.)

In the first rotation, the subtree with root X is rotated right insuch a way so that the right child of Y now points to X.

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Session 16Ver. 1.0

P

Yr

Pl

X

Y

Xr

-1

Yl

1

-2

In the second rotation, the subtree with root P is rotated leftso that Y is moved upwards to form the root.

The left child of Y now becomes the right child of P.

P

-2

Yl

Yr

Pl

X-1

-1

Y

Xr

After single right rotation

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Session 16Ver. 1.0

In the second rotation, the subtree with root P is rotated leftso that Y is moved upwards to form the root.

The left child of Y now points to P.

P

-2

Yl

Yr

Pl

X-1

-1

Y

Xr

P

0

Yl YrPl

X-1

0

Y

Xr

After single left rotation

The tree is now balanced.

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

Let us now consider another case in which the pivot node isleft heavy and the new node is inserted in the left subtree ofthe pivot node.

This further involves two cases:

If a new node is inserted in the left subtree of the left child of

the pivot node.If the new node is inserted in the right subtree of the left childof the pivot node.

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

Let us first consider the case in which the new node isinserted in the left subtree of the left child of the pivot node.

Initially, the binary tree appears as:

P

X

0

1

Xr

Pr

Xl

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

When a new node is inserted in the left subtree of the leftchild of the pivot node (P), the tree becomes unbalanced.

P

X

0

1

Xr

Pr

Xl

1

2

Inserting Nodes in a Height Balanced Tree (Contd.)

Xl

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Session 16Ver. 1.0

To restore the balance, you need to perform a single rightrotation.

P

X

Xr

Pr

2

1

Inserting Nodes in a Height Balanced Tree (Contd.)

Xl

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Session 16Ver. 1.0

P

X

Xr

Pr

Xl

2

1

P

X

0

0

Xr Pr

Xl

After single right rotation

The tree is now balanced.

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

Let us now consider a case in which the new node is insertedin the right subtree of the left child of the pivot node.

Initially, the tree appears as:

P

X

1

Yl Yr

Xl

YPr

0

0

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

When a new node is inserted in the right subtree of the leftchild of pivot node, P, the tree becomes unbalanced.

P

X0

1

Yl

Xl

YPr

0

YrYr

2

-1

-1

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

In this case, a double rotation is required to restore thebalance.

P

X

Yl

Xl

YPr

Yr

2

-1

-1

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

A left rotation is performed first in which the subtree withroot X is rotated towards left.

P

X

Yl

Xl

YPr

Yr

2

-1

-1

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

A left rotation is performed first in which the subtree withroot X is rotated towards left.

P

X

Yl

Xl

YPr

Yr

2

-1

-1

P

X

-1

2

Yl

Yr

Xl

Y

Pr

-1

After single left rotation

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

In the next step, a single right rotation is performed in whichthe subtree with root P is rotated right.

P

X

Yl

Xl

YPr

Yr

2

-1

-1

P

X

-1

2

Yl

Yr

Xl

Y

Pr

-1

After single left rotation

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

In the next step, a single right rotation is performed in whichthe subtree with root P is rotated right.

P

X

-1

2

Yl

Yr

Xl

Y

Pr

-1

After single right rotation

PX

0

0

YlYrXl

Y

1

Pr

The tree is now balanced.

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

Let us consider an example to insert values in a binarysearch tree and restore its balance whenever required.50 40 30 60 55 80 10 35 32

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

Insert 50

0

50

Tree is balanced

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

Insert 40

0

50

Tree is balanced

0

40

1

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

Insert 30

50

Tree is balanced

0

40

1

0

30

2

1

Tree becomes

unbalanced

A single right rotation restores the balance

Inserting Nodes in a Height Balanced Tree (Contd.)

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Insert 30

50

40

030

0

0

Tree is now

balanced

Inserting Nodes in a Height Balanced Tree (Contd.)

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Insert 60

50

40

030

0

0

Tree is now

balanced

0

60

-1

-1

Tree is balanced

Inserting Nodes in a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

Insert 55

50

40

030

0

60

Tree is balanced

-1

-1

0

55

-2

-2

1

Tree becomes

unbalanced

A double rotation restores the balance

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

i d i i h l d ( d )

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Insert 55

50

40

030

60

0

55

A double rotation restores the balance

Tree becomes

unbalanced

1

-2

-2

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

I i N d i H i h B l d T (C d )

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

50

40

030

55

0

60

A double rotation restores the balance

Tree becomes

unbalanced

-1

-2

-2

60

0

55

1

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

I ti N d i H i ht B l d T (C td )

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

55

40

030

60

0

50

A double rotation restores the balance

Tree becomes

unbalanced

-1

-2

-2

0

0

-1

0

60

Tree is now

balanced

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

I ti N d i H i ht B l d T (C td )

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Session 16Ver. 1.0

50 40 30 60 55 80 10 35 32

Insert 80

55

40

030

60

0

50

Tree is now

balanced

0

0

-1

80

0

-1

-1

Tree becomes

unbalanced

A single left rotation restores the balance

Inserting Nodes in a Height Balanced Tree (Contd.)

-2

Data Structures and Algorithms

I ti N d i H i ht B l d T (C td )

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50 40 30 60 55 80 10 35 32

60

55

040

8050

0

-1

0

30

0 0

A single left rotation restores the balance

Tree is now

balanced

55

40

030

60

0

50

80

0

-1

-1

Inserting Nodes in a Height Balanced Tree (Contd.)

-2

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Data Structures and Algorithms

Inse ting Nodes in a Height Balanced T ee (Contd )

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Session 16Ver. 1.0

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60

55

40

8050

0

-1

30

0

Insert 35

10

0

Tree is balanced

1

1

1

35

0

0

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

Inserting Nodes in a Height Balanced Tree (Contd )

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50 40 30 60 55 80 10 35 32

60

55

40

8050

0

-1

30

0

Insert 32

10

0

Tree is balanced

1

1

35

0

0

32

0

1

-1

2

2

Tree is becomes

unbalanced

A double rotation is required torestore the balance

Inserting Nodes in a Height Balanced Tree (Contd.)

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Inserting Nodes in a Height Balanced Tree (Contd )

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60

55

40

8050

0

-1

30

0

Insert 32

10

0

35

32

0

A double rotation is required torestore the balance

1

-1

2

2

Tree is becomesunbalanced

Inserting Nodes in a Height Balanced Tree (Contd.)

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50 40 30 60 55 80 10 35 32

60

55

40

8050

0

-1

35

0

30

0

35

32

0

A double rotation is required torestore the balance

1

-1

2

2

10

0

2

Inserting Nodes in a Height Balanced Tree (Contd.)

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50 40 30 60 55 80 10 35 32

60

55

35

8040

0

-1

30

-1

10

0

32

A double rotation is required torestore the balance

0

0

0

50

00

Tree is nowbalanced

Inserting Nodes in a Height Balanced Tree (Contd.)

Data Structures and Algorithms

Just a minute

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Session 16Ver. 1.0

In which situations do you need to implement a doublerotation to balance the tree?

Just a minute

Answer:

There can be two situations where a double rotation is requiredto balance the tree:

If the pivot node is right heavy and the right child of pivot node isleft heavy

If the pivot node is left heavy and the left child of pivot node isright heavy

Data Structures and Algorithms

Deleting Nodes from a Height Balanced Tree

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Deleting Nodes from a Height Balanced Tree

Delete operation in a height balanced tree is same as that ina normal binary tree.

However, if deletion of a node makes the tree unbalanced,then you need to perform appropriate rotations to restorethe balance.

This process involves the same concept as used in insertionof a node.

Data Structures and Algorithms

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Session 16Ver. 1.0

Consider the following height balanced binary tree.

55

40 70

30 50 60 80

10 35 53 90

32

1

1 -1

-1 -1

0 1

0

0

0 -1

0

Deleting Nodes from a Height Balanced Tree (Contd.)

Data Structures and Algorithms

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Session 16Ver. 1.0

Let us implement a few delete operations.

55

40 70

30 50 60 80

10 35 53 90

32

1

1 -1

-1 -1

0 1

0

0

0 -1

0

Deleting Nodes from a Height Balanced Tree (Contd.)

Data Structures and Algorithms

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Session 16Ver. 1.0

55

40 70

30 50 60 80

10 35 53 90

32

1

1 -1

-1 -1

0 1

0

0

0 -1

0

2

0

Deleting Nodes from a Height Balanced Tree (Contd.)

Delete 53

Data Structures and Algorithms

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Session 16Ver. 1.0

55

40 70

30 50 60 80

10 35 90

32

1

-1

-1

0 1

0

0 -1

0

2

0

Tree becomes unbalanced

A double rotation is required to restore the balance

Deleting Nodes from a Height Balanced Tree (Contd.)

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55

40 70

30 50 60 80

10 35 90

32

1

-1

-1

0 1

0

0 -1

0

2

0

Tree becomes unbalanced

Rotate left around node 30

Deleting Nodes from a Height Balanced Tree (Contd.)

Data Structures and Algorithms

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55

40 70

35 50 60 80

30 90

32

1

-1

2

0

0

0 -1

0

2

0

Rotate right around node 40

10

0

Deleting Nodes from a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

55

35 70

40 60 8030

9032

0

-1

0

0

0 -1

0

0

-1

Rotate right around node 40

10

0

50

0

Tree is now balanced

Deleting Nodes from a Height Balanced Tree (Contd.)

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Session 16Ver. 1.0

55

35 70

40 60 8030

9032

0

-1

0

0

0 -1

0

0

-1

10

0

50

0

Tree becomes unbalancedDelete 60

-2

A single left rotation restores the balance

Deleting Nodes from a Height Balanced Tree (Contd.)

Data Structures and Algorithms

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Session 16Ver. 1.0

55

35 80

40 9030 70

32

1

0

0

00

0

-1

10

0

50

0

0

Tree becomes balanced

Deleting Nodes from a Height Balanced Tree (Contd.)

Data Structures and Algorithms

Deleting Nodes from a Height Balanced Tree (Contd.)

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