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Data Structures and Algorithms
Session 16Ver. 1.0
Objectives
In this session, you will learn to:Implement a threaded binary tree
Implement a height balanced binary tree
Store data in a graph
Data Structures and Algorithms
Session 16Ver. 1.0
Deleting a Node from a Threaded Binary Tree
Delete operation in a threaded binary tree refers to the process of removing the specified node from the threaded binary 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 marked as the currentNode and its parent as parent.
Write an algorithm to locate the node to be deleted in a threaded binary tree.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
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40. ..
65.
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header node
. .
.
Algorithm to locate the node to be deleted and its parent in a threaded binary tree.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Delete 80
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete 80
currentNode
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 node
to be searched is found or currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 node
to be searched is found or currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 node
to be searched is found or currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
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 currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 node
to be searched is found or currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 node
to be searched is found or currentNode becomes NULL:
a. Mark currentNode as parent.b. If the value to be searched is less
than that of currentNode, and the left child 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 left child 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 the right child of the currentNode is a link:
i. Make currentNode point to its right child and go to step 3.
e. If the value to be searched is greater than that of currentNode, and the right child of the currentNode is a thread:
i. Mark currentNode as NULL and go to step 3.
Data Structures and Algorithms
Session 16Ver. 1.0
Once you locate the node to be deleted and its parent, you can release the memory of the node to be deleted after adjusting the links and threads appropriately.
Before implementing a delete operation, you first need to check whether the tree is empty or not.
The tree is empty if the left child of the header node is a thread 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.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us first consider a case in which the node to be deleted is the leaf node.
In this case, you first need to check if there is only one node present in the tree.
header node
65
new node
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
To delete this node, make the left child of the header node as a thread pointing to itself.
header node
65
new node
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
To delete this node, make the left child of the header node as 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.)
Data Structures and Algorithms
Session 16Ver. 1.0
However, if there are more than one nodes in the tree, you need another algorithm to delete a leaf node.
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
30 50 80
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72
69
40. ..
65.
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header node
. .
.
Delete node 60
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 60
currentNode
parent
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 60
currentNode
parent
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 60
currentNode
parent
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 60
currentNode
parent
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 60
currentNode
parent
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
.
.
Delete node 60
currentNode
parent
Deletion complete
1. Locate the position of the node to be deleted. Mark it as currentNode and its parent as parent.
2. If currentNode is the left child of parent:
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 predecessor of currentNode.
c. Go to step 4.
3. If currentNode is the right child of parent:
a. Set the right thread field of parent as zero.
b. Make the right child field of parent as a thread pointing to the inorder successor of currentNode.
4. Release the memory for currentNode.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Write an algorithm to delete a node, which has one child in a threaded binary tree.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us now consider a case in which the node to be deleted has one child (left or right).
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Algorithm to delete a node with one child from a threaded binary tree.
Delete node 50
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.parent
currentNode
Delete node 50
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
Delete node 50
parent
currentNode
child
Deleting a Node from a Threaded Binary Tree (Contd.) 1. Locate the node to be deleted. Mark it as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
60
72
69
40. ..
65.
.
header node
. .
.
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 as
currentNode 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 inorder
predecessor of currentNode. Mark them as successor 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 left thread of successor to zero.
b. Go to step 9.8. If currentNode has a left child:
a. Make the right child of its predecessor point to its successor and set the right thread of predecessor to zero.
9. Release the memory for currentNode.
Data Structures and Algorithms
Session 16Ver. 1.0
Write an algorithm to delete a node, which has two children from a threaded binary tree.Algorithm to delete a node having two children from a threaded binary tree.
1. Locate the position of the node to be deleted. Mark it as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
40. ..
65.
.
header node
.
.
Delete node 40 1. Locate the position of the node to be deleted. Mark it as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Deleting a Node from a Threaded Binary Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
40. ..
65.
.
header node
.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
40. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
40. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
45. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
45. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
45. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
30 50 80
45
72
69
45. ..
65.
.
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 as currentNode.
2. Locate the inorder successor of currentNode. Mark it as Inorder_suc and its parent as Inorder_parent.
3. Overwrite the data contained in currentNode with the data contained in Inorder_suc.
4. Delete the node marked Inorder_suc. This node would have at most one child and can therefore be deleted by using the algorithm for one of the following:
a. Deleting a leaf node.
b. Deleting a node with one child.
Data Structures and Algorithms
Session 16Ver. 1.0
Implementing a Height Balanced Tree
In a binary search tree, the time required to search for a particular value depends upon its height.
The shorter the height, the faster is the search.
However, binary search trees tend to attain large heights because of continuous insert and delete operations.
Data Structures and Algorithms
Session 16Ver. 1.0
After inserting values in the specified order, the binary search tree appears as follows:Consider an example in which you want to insert some numeric 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
Data Structures and Algorithms
Session 16Ver. 1.0
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 binary search tree, you will have to traverse all its preceding nodes before you reach node 14. In this case, you need to make 14 comparisons.
In such a case, the binary search tree becomes equivalent to a linked list.
Defining Height Balanced Tree (Contd.)
This process can be very time consuming if the number of values stored in a binary search tree is large.
Data Structures and Algorithms
Session 16Ver. 1.0
Therefore, such a structure loses its property of a binary search tree in which after every comparison, the search operations are reduced to half.
12
34
56
78
910
1112
1314
15
Defining Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
To solve this problem, it is desirable to keep the height of the tree to a minimum.
Therefore, the following binary search tree can be modified to reduce its height.
12
34
56
78
910
1112
1314
15
Defining Height Balanced Tree (Contd.)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Data Structures and Algorithms
Session 16Ver. 1.0
The height of the binary search tree has now been reduced to 4.Now if you want to search for a value 14, you just need to traverse nodes 8 and 12, before you reach node 14.In this case, the total number of comparisons to be made to search for node 14 is three.
This approach reduces the time to search for a particular value in a binary search tree.
Defining Height Balanced Tree (Contd.)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Data Structures and Algorithms
Session 16Ver. 1.0
This can be implemented with the help of a height balanced tree.
Defining Height Balanced Tree (Contd.)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Data Structures and Algorithms
Session 16Ver. 1.0
A height balanced tree is a binary tree in which the difference between the heights of the left subtree and right subtree 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 of a node is equal to the height of its right subtree.
1: A BF value of 1 indicates that the height of the left subtree is greater 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 right subtree is greater than the height of the left subtree by one. A node in this state is said to be right heavy.
Defining Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50
40
0
30
0
60
-1
-1
50
40
0
30
0
0
50
40
1
30
0
1
0
10
Balanced Binary Search Trees
Defining Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
After inserting a new node in a height balanced tree, the balance factor of one or more nodes may attain a value other 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 inserted node, which has a balance factor other than 1, 0 or –1.
To restore the balance, you need to perform appropriate rotations around the pivot node.
Defining Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50
40
0
30
1
60
-2
-2
50
40
2
30
0
1
55
0
35
1
34
055
0
45
0
Unbalanced Binary Search Trees
Defining Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
In a height balanced tree, the maximum difference between the height of left and right subtree of a node can be _________.
Just a minute
Answer:One
Data Structures and Algorithms
Session 16Ver. 1.0
Inserting Nodes in a Height Balanced Tree
Insert operation in a height balanced tree is similar to that in a simple binary search tree.
However, inserting a node can make the tree unbalanced.
To restore the balance, you need to perform appropriate rotations around the pivot node.
This involves two cases:When the pivot node is initially right heavy and the new node is inserted in the right subtree of the pivot node.
When the pivot node is initially left heavy and the new node is inserted in the left subtree of the pivot node.
Data Structures and Algorithms
Session 16Ver. 1.0
Let us first consider a case in which the pivot node is initially right heavy and the new node is inserted in the right subtree of the pivot node.
In this case, after the insertion of a new node, the balance factor 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 of the pivot node.
If the new node is inserted in the left subtree of the right child of the pivot node.
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Consider the first case in which a new node is inserted in the right subtree of the right child of the pivot node.
In this case, the binary tree initially appears as:
P
X
0
Xl Xr
Pl
-1
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
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
Data Structures and Algorithms
Session 16Ver. 1.0
To restore the balance, you need to perform a single left rotation 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
Data Structures and Algorithms
Session 16Ver. 1.0
In the second case, a new node is inserted in the left subtree of X.
In this case, the binary tree initially appears as:
P
-1
Yl Yr
Pl
X
0
0
Y
Xr
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
When a new node is inserted in the left subtree of X, the binary tree becomes unbalanced.
P
-1
Yl Yr
Pl
X0
Y
Xr
Yl
-2
-10
1
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
To restore the balance, you need to perform a double rotation.In the first rotation, the subtree with root X is rotated right in such 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
Data Structures and Algorithms
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 in such a way so that the right child of Y now points to X.
Data Structures and Algorithms
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 left so 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
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
In the second rotation, the subtree with root P is rotated left so 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 Yr
Pl
X-1
0
Y
Xr
After single left rotation
The tree is now balanced.
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us now consider another case in which the pivot node is left heavy and the new node is inserted in the left subtree of the 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 child of the pivot node.
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us first consider the case in which the new node is inserted 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.)
Data Structures and Algorithms
Session 16Ver. 1.0
When a new node is inserted in the left subtree of the left child 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
Data Structures and Algorithms
Session 16Ver. 1.0
To restore the balance, you need to perform a single right rotation.
P
X
Xr
Pr
2
1
Inserting Nodes in a Height Balanced Tree (Contd.)
Xl
Data Structures and Algorithms
Session 16Ver. 1.0
P
X
Xr
Pr
Xl
2
1P
X
0
0
Xr Pr
Xl
After single right rotation
The tree is now balanced.
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us now consider a case in which the new node is inserted in the right subtree of the left child of the pivot node.
Initially, the tree appears as:
P
X
1
Yl YrXl
YPr
0
0
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
When a new node is inserted in the right subtree of the left child of pivot node, P, the tree becomes unbalanced.
P
X0
1
YlXl
YPr
0
YrYr
2
-1
-1
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
In this case, a double rotation is required to restore the balance.
P
X
YlXl
YPr
Yr
2
-1
-1
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
A left rotation is performed first in which the subtree with root X is rotated towards left.
P
X
YlXl
YPr
Yr
2
-1
-1
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
A left rotation is performed first in which the subtree with root X is rotated towards left.
P
X
YlXl
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.)
Data Structures and Algorithms
Session 16Ver. 1.0
In the next step, a single right rotation is performed in which the subtree with root P is rotated right.
P
X
YlXl
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.)
Data Structures and Algorithms
Session 16Ver. 1.0
In the next step, a single right rotation is performed in which the 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.)
Data Structures and Algorithms
Session 16Ver. 1.0
Let us consider an example to insert values in a binary search tree and restore its balance whenever required.50 40 30 60 55 80 10 35 32
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
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.)
Data Structures and Algorithms
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.)
Data Structures and Algorithms
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.)
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
Insert 30
50
40
0
30
0
0
Tree is now balanced
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
Insert 60
50
40
0
30
0
0
Tree is now balanced
0
60
-1
-1
Tree is balanced
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
Insert 55
50
40
0
30
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
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
Insert 55
50
40
0
30
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
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
50
40
0
30
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
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
55
40
0
30
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
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
Insert 80
55
40
0
30
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
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
60
55
0
40
8050
0
-1
0
30
0 0
A single left rotation restores the balance
Tree is now balanced
55
40
0
30
60
0
50
80
0
-1
-1
Inserting Nodes in a Height Balanced Tree (Contd.)
-2
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
60
55
0
40
8050
0
-1
0
30
0 0
Insert 10
10
0
1
1
1
Tree is balanced
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
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
Session 16Ver. 1.0
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 to restore the balance
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
50 40 30 60 55 80 10 35 32
60
55
40
8050
0
-1
30
0
Insert 32
10
0
35
32
0
A double rotation is required to restore the balance
1
-1
2
2
Tree is becomes unbalanced
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
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 to restore the balance
1
-1
2
2
10
0
2
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
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 to restore the balance
0
0
0
50
00
Tree is now balanced
Inserting Nodes in a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
In which situations do you need to implement a double rotation to balance the tree?
Just a minute
Answer:There can be two situations where a double rotation is required to balance the tree:
If the pivot node is right heavy and the right child of pivot node is left heavy
If the pivot node is left heavy and the left child of pivot node is right heavy
Data Structures and Algorithms
Session 16Ver. 1.0
Deleting Nodes from a Height Balanced Tree
Delete operation in a height balanced tree is same as that in a normal binary tree.
However, if deletion of a node makes the tree unbalanced, then you need to perform appropriate rotations to restore the balance.
This process involves the same concept as used in insertion of a node.
Data Structures and Algorithms
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
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
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
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.)
Data Structures and Algorithms
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
Rotate left around node 30
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
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.)
Data Structures and Algorithms
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.)
Data Structures and Algorithms
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
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
Session 16Ver. 1.0
55
35 80
40 9030 70
32
1
0
0
00
0
-1
10
0
50
0
0
Delete 50
0
1
Tree is balanced
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
55
35 80
40 9030 70
32
1
0
0
00
10
0
0
Delete 40 Tree is balanced
0
12
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
55
35 80
9030 70
32
1
0
0
00
10
0
0
Delete 40 Tree becomes unbalanced
2
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
55
35 80
9030 70
32
1
0
0
00
10
0
0
Tree becomes unbalanced
2
A single right rotation is required to restore the balance
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
55
30 80
9010 70
32
1
0
0
00
0
Tree is now balanced
35
-1
1
A single right rotation is required to restore the balance
Deleting Nodes from a Height Balanced Tree (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
Consider a situation:You have to visit a set of cities and return back to the original city in the end.
For this, you need to:Find the shortest or least expensive path that starts from the original city, visits each of the desired cities, and then returns back to the original city.
How will you solve this problem?
Storing Data in a Graph
Data Structures and Algorithms
Session 16Ver. 1.0
To solve this problem, you need to:Determine a way of representing the information, pertaining to the various cities and the distances between the pairs of cities.
Such pair-wise relationships can be represented in the form of a graph.
Storing Data in a Graph (Contd.)
Data Structures and Algorithms
Session 16Ver. 1.0
A graph is defined as a data structure that consists of a set of vertices (nodes) and a set of edges (arcs).
It is a collection of data items that are interconnected to form a network.
There are two types of graphs:Directed graph
Undirected graph
Defining Graph
Data Structures and Algorithms
Session 16Ver. 1.0
Just a minute
A graph in which any node in a graph is adjacent to all the other nodes in a graph is called a __________.
1. Undirected graph
2. Directed graph
3. Complete graph
4. Cyclic graph
Answer: 3. Complete graph
Data Structures and Algorithms
Session 16Ver. 1.0
Consider the given graph.
Identify the type of graph.
Just a minute
Answer:The given graph is a directed graph.
Data Structures and Algorithms
Session 16Ver. 1.0
Summary
In this session, you learned that:A height balanced tree is a binary tree in which the difference between the height of the left and right subtree is not more than one.
A tree becomes unbalanced if any node in the tree attains a balance factor other than 0, 1, or –1.
If the tree becomes unbalanced, you need to balance the tree by performing appropriate rotations around the pivot node.
A graph is defined as a data structure that consists of a set of vertices (nodes) and edges (arcs).
There are two types of graphs:Directed
Undirected
