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<!DOCTYPE html>

<head>

window.dataLayer = window.dataLayer || [];

function gtag() { dataLayer.push(arguments); }

gtag('js', new Date());

gtag('config', 'G-HCERBDV76D');

</script>

</head>

<body>

<nav>

Python Tutorial

</a>

</div>

</div>

</div>

<li>

Home

</a>

</li>

<li>

Projects

</a>

</li>

<li>

Free Course

</a>

</li>

<li>

DSA

</a>

</li>

</ul>

</div>

</nav>

</div>

<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

In this tutorial, you will learn what an avl tree is. Also, you will find working examples of various operations performed on an avl tree in C, C++, Java and Python.

AVL tree is a self-balancing binary search tree in which each node maintains extra information called a balance factor whose value is either -1, 0 or +1.

AVL tree got its name after its inventor Georgy Adelson-Velsky and Landis.

<hr><h2 id="balancefactor">Balance Factor</h2>

Balance factor of a node in an AVL tree is the difference between the height of the left subtree and that of the right subtree of that node.

Balance Factor = (Height of Left Subtree - Height of Right Subtree) or (Height of Right Subtree - Height of Left Subtree)

The self balancing property of an avl tree is maintained by the balance factor. The value of balance factor should always be -1, 0 or +1.

An example of a balanced avl tree is:

<figure><img alt="avl tree" src="//cdn.programiz.com/sites/tutorial2program/files/avl-tree-final-tree-1_0_2.png" title="avl tree final" width="300" height="260"><figcaption>Avl tree</figcaption></figure><hr><h2>Operations on an AVL tree</h2>

Various operations that can be performed on an AVL tree are:

<h2 id="rotations">Rotating the subtrees in an AVL Tree</h2>

In rotation operation, the positions of the nodes of a subtree are interchanged.

There are two types of rotations:

<hr><h3>Left Rotate</h3>

<div class="clearfix"></div>In left-rotation, the arrangement of the nodes on the right is transformed into the arrangements on the left node.

Algorithm

<ol><li>Let the initial tree be:

<figure><img alt="left-rotate" src="//cdn.programiz.com/sites/tutorial2program/files/avl-tree_leftrotate-1.png" title="left rotate" width="176" height="208"><figcaption>Left rotate</figcaption></figure></li>

<li>If <var>y</var> has a left subtree, assign <var>x</var> as the parent of the left subtree of <var>y</var>.

<figure><img alt="left-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_leftrotate-2.png" title="assign x as the parent of the left subtree of y" width="230" height="209"><figcaption>Assign x as the parent of the left subtree of y</figcaption></figure></li>

<li>If the parent of <var>x</var> is <code>NULL</code>, make <var>y</var> as the root of the tree.</li>

<li>Else if <var>x</var> is the left child of <var>p</var>, make <var>y</var> as the left child of <var>p</var>.</li>

<li>Else assign <var>y</var> as the right child of <var>p</var>.

<figure><img alt="left-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_leftrotate-3.png" title="change the parent of x to that of y" width="200" height="156"><figcaption>Change the parent of x to that of y</figcaption></figure></li>

<li>Make <var>y</var> as the parent of <var>x</var>.

<figure><img alt="left-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_leftrotate-4.png" title="Assign y as the parent of x." width="200" height="223"><figcaption>Assign y as the parent of x.</figcaption></figure></li>

</ol><hr><h3>Right Rotate</h3>

In left-rotation, the arrangement of the nodes on the left is transformed into the arrangements on the right node.

<ol><li>Let the initial tree be:

<figure><img alt="right-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_rightrotate-1.png" title="initial tree" width="200" height="224"><figcaption>Initial tree</figcaption></figure></li>

<li>If <var>x</var> has a right subtree, assign y as the parent of the right subtree of <var>x</var>.

<figure><img alt="right-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_rightrotate-2.png" title="assign y as the parent of the right subtree of x." width="230" height="215"><figcaption>Assign y as the parent of the right subtree of x</figcaption></figure></li>

<li>If the parent of <var>y</var> is <code>NULL</code>, make <var>x</var> as the root of the tree.</li>

<li>Else if <var>y</var> is the right child of its parent <var>p</var>, make <var>x</var> as the right child of <var>p</var>.</li>

<li>Else assign <var>x</var> as the left child of <var>p</var>.

<figure><img alt="right-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_rightrotate-3.png" title="Assign the parent of y as the parent of x." width="200" height="161"><figcaption>Assign the parent of y as the parent of x.</figcaption></figure></li>

<li>Make <var>x</var> as the parent of <var>y</var>.

<figure><img alt="right-rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_rightrotate-4.png" title="assign x as the parent of y" width="200" height="224"><figcaption>Assign x as the parent of y</figcaption></figure></li>

</ol><hr><h3>Left-Right and Right-Left Rotate</h3>

In left-right rotation, the arrangements are first shifted to the left and then to the right.

<ol><li>Do left rotation on x-y.

<figure><img alt="left-right rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree-leftright-rotate-1.png" title="left rotate x-y" width="450" height="247"><figcaption>Left rotate x-y</figcaption></figure></li>

<li>Do right rotation on y-z.

<figure><img alt="left-right rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree-leftright-rotate-2.png" title="right rotate z-y" width="450" height="233"><figcaption>Right rotate z-y</figcaption></figure></li>

</ol>In right-left rotation, the arrangements are first shifted to the right and then to the left.

<ol><li>Do right rotation on x-y.

<figure><img alt="right-left rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree-rightleft-rotate-1.png" title="right rotate x-y" width="450" height="240"><figcaption>Right rotate x-y</figcaption></figure></li>

<li>Do left rotation on z-y.

<figure><img alt="right-left rotate" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree-rightleft-rotate-2.png" title="left rotate z-y" width="450" height="235"><figcaption>Left rotate z-y</figcaption></figure></li>

</ol><hr><h2 id="insertion">Algorithm to insert a newNode</h2>

A <var>newNode</var> is always inserted as a leaf node with balance factor equal to 0.

<ol><li>Let the initial tree be:

<figure><img alt="initial tree" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_initial-tree.png" title="initial tree for insertion" width="450" height="325"><figcaption>Initial tree for insertion</figcaption></figure>

Let the node to be inserted be:

<li>Go to the appropriate leaf node to insert a <var>newNode</var> using the following recursive steps. Compare <var>newKey</var> with <var>rootKey</var> of the current tree.

<ol type="a"><li>If <var>newKey</var> < <var>rootKey</var>, call insertion algorithm on the left subtree of the current node until the leaf node is reached.</li>

<li>Else if <var>newKey</var> > <var>rootKey</var>, call insertion algorithm on the right subtree of current node until the leaf node is reached.</li>

<li>Else, return <var>leafNode</var>.

<figure><img alt="avl tree insertion" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_find-position.png" title="finding the location to insert newNode" width="500" height="275"><figcaption>Finding the location to insert newNode</figcaption></figure></li>

</ol></li>

<li>Compare <var>leafKey</var> obtained from the above steps with <var>newKey</var>:

<ol type="a"><li>If <var>newKey</var> < <var>leafKey</var>, make newNode as the <var>leftChild</var> of <var>leafNode</var>.</li>

<li>Else, make <var>newNode</var> as rightChild of <var>leafNode</var>.

<figure><img alt="avl tree insertion" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_insert.png" title="inserting the new node" width="450" height="364"><figcaption>Inserting the new node</figcaption></figure></li>

</ol></li>

<li>Update <var>balanceFactor</var> of the nodes.

<figure><img alt="avl tree insertion" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_update-bf.png" title="updating the balance factor after insertion" width="400" height="355"><figcaption>Updating the balance factor after insertion</figcaption></figure></li>

<li>If the nodes are unbalanced, then rebalance the node.

<ol type="a"><li>If <var>balanceFactor</var> > 1, it means the height of the left subtree is greater than that of the right subtree. So, do a right rotation or left-right rotation

<ol type="i"><li>If <var>newNodeKey</var> < <var>leftChildKey</var> do right rotation.</li>

<li>Else, do left-right rotation.

<figure><img alt="insertion in avl tree" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_balance1.png" title="balancing the tree with rotation" width="1200" height="460"><figcaption>Balancing the tree with rotation</figcaption></figure><figure><img alt="insertion in avl tree" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_balnce2.png" title="balancing the tree with rotation" width="1200" height="468"><figcaption>Balancing the tree with rotation</figcaption></figure></li>

</ol></li>

<li>If <var>balanceFactor</var> < -1, it means the height of the right subtree is greater than that of the left subtree. So, do right rotation or right-left rotation

<ol type="i"><li>If <var>newNodeKey</var> > <var>rightChildKey</var> do left rotation.</li>

<li>Else, do right-left rotation</li>

</ol></li>

<li>The final tree is:

<figure><img alt="left-right insertion" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_final-tree.png" title="final balanced tree" width="400" height="293"><figcaption>Final balanced tree</figcaption></figure></li>

</ol><hr><h2 id="deletion">Algorithm to Delete a node</h2>

A node is always deleted as a leaf node. After deleting a node, the balance factors of the nodes get changed. In order to rebalance the balance factor, suitable rotations are performed.

<ol><li>Locate <var>nodeToBeDeleted</var> (recursion is used to find <var>nodeToBeDeleted</var> in the code used below).

<figure><img alt="node to be deleted" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_nodeToBeDeleted_0.png" title="locating the node to be deleted" width="400" height="293"><figcaption>Locating the node to be deleted</figcaption></figure></li>

<li>There are three cases for deleting a node:

<ol type="a"><li>If <var>nodeToBeDeleted</var> is the leaf node (ie. does not have any child), then remove <var>nodeToBeDeleted</var>.</li>

<li>If <var>nodeToBeDeleted</var> has one child, then substitute the contents of <var>nodeToBeDeleted</var> with that of the child. Remove the child.</li>

<li>If <var>nodeToBeDeleted</var> has two children, find the inorder successor <var>w</var> of <var>nodeToBeDeleted</var> (ie. node with a minimum value of key in the right subtree).

<figure><img alt="finding the successor" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_successor_0.png" title="finding the successor" width="400" height="293"><figcaption>Finding the successor</figcaption></figure><ol type="i"><li>Substitute the contents of <var>nodeToBeDeleted</var> with that of <var>w</var>.

<figure><img alt="substitute the node to be deleted" src="https://cdn.programiz.com/sites/tutorial2program/files/avl-tree_substitute_0.png" title="substitute the node to be deleted" width="400" height="258"><figcaption>Substitute the node to be deleted</figcaption></figure></li>

<li>Remove the leaf node <var>w</var>.

<figure><img alt="remove w" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_removal_0.png" title="remove w" width="400" height="293"><figcaption>Remove w</figcaption></figure></li>

</ol></li>

<li>Update <var>balanceFactor</var> of the nodes.

<figure><img alt="update bf" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree_upatebf_0.png" title="update bf" width="400" height="293"><figcaption>Update bf</figcaption></figure></li>

<li>Rebalance the tree if the balance factor of any of the nodes is not equal to -1, 0 or 1.

<ol type="a"><li>If <var>balanceFactor</var> of <var>currentNode</var> > 1,

<ol type="i"><li>If <var>balanceFactor</var> of <var>leftChild</var> >= 0, do right rotation.

<figure><img alt="right-rotate" src="/sites/tutorial2program/files/avl-tree_balance_0.png" title="right-rotate for balancing the tree" width="700" height="262"><figcaption>Right-rotate for balancing the tree</figcaption></figure></li>

<li>Else do left-right rotation.</li>

</ol></li>

<li>If <var>balanceFactor</var> of <var>currentNode</var> < -1,

<ol type="i"><li>If <var>balanceFactor</var> of <var>rightChild</var> <= 0, do left rotation.</li>

<li>Else do right-left rotation.</li>

</ol></li>

<li>The final tree is:

<figure><img alt="avl tree" src="https://www.programiz.com/sites/tutorial2program/files/avl-tree-final-tree-1_0_2.png" title="avl tree final" width="300" height="260"><figcaption>Avl tree final</figcaption></figure></li>

</ol><hr><h2 id="code">Python, Java and C/C++ Examples</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"># AVL tree implementation in Python

import sys

# Create a tree node

class TreeNode(object):

def __init__(self, key):

self.key = key

self.left = None

self.right = None

self.height = 1

class AVLTree(object):

# Function to insert a node

def insert_node(self, root, key):

# Find the correct location and insert the node

if not root:

return TreeNode(key)

elif key < root.key:

root.left = self.insert_node(root.left, key)

else:

root.right = self.insert_node(root.right, key)

root.height = 1 + max(self.getHeight(root.left),

self.getHeight(root.right))

# Update the balance factor and balance the tree

balanceFactor = self.getBalance(root)

if balanceFactor > 1:

if key < root.left.key:

return self.rightRotate(root)

else:

root.left = self.leftRotate(root.left)

return self.rightRotate(root)

if balanceFactor < -1:

if key > root.right.key:

return self.leftRotate(root)

else:

root.right = self.rightRotate(root.right)

return self.leftRotate(root)

return root

# Function to delete a node

def delete_node(self, root, key):

# Find the node to be deleted and remove it

if not root:

return root

elif key < root.key:

root.left = self.delete_node(root.left, key)

elif key > root.key:

root.right = self.delete_node(root.right, key)

else:

if root.left is None:

temp = root.right

root = None

return temp

elif root.right is None:

temp = root.left

root = None

return temp

temp = self.getMinValueNode(root.right)

root.key = temp.key

root.right = self.delete_node(root.right,

temp.key)

if root is None:

return root

# Update the balance factor of nodes

root.height = 1 + max(self.getHeight(root.left),

self.getHeight(root.right))

balanceFactor = self.getBalance(root)

# Balance the tree

if balanceFactor > 1:

if self.getBalance(root.left) >= 0:

return self.rightRotate(root)

else:

root.left = self.leftRotate(root.left)

return self.rightRotate(root)

if balanceFactor < -1:

if self.getBalance(root.right) <= 0:

return self.leftRotate(root)

else:

root.right = self.rightRotate(root.right)

return self.leftRotate(root)

return root

# Function to perform left rotation

def leftRotate(self, z):

y = z.right

T2 = y.left

y.left = z

z.right = T2

z.height = 1 + max(self.getHeight(z.left),

self.getHeight(z.right))

y.height = 1 + max(self.getHeight(y.left),

self.getHeight(y.right))

return y

# Function to perform right rotation

def rightRotate(self, z):

y = z.left

T3 = y.right

y.right = z

z.left = T3

z.height = 1 + max(self.getHeight(z.left),

self.getHeight(z.right))

y.height = 1 + max(self.getHeight(y.left),

self.getHeight(y.right))

return y

# Get the height of the node

def getHeight(self, root):

if not root:

return 0

return root.height

# Get balance factore of the node

def getBalance(self, root):

if not root:

return 0

return self.getHeight(root.left) - self.getHeight(root.right)

def getMinValueNode(self, root):

if root is None or root.left is None:

return root

return self.getMinValueNode(root.left)

def preOrder(self, root):

if not root:

return

print("{0} ".format(root.key), end="")

self.preOrder(root.left)

self.preOrder(root.right)

# Print the tree

def printHelper(self, currPtr, indent, last):

if currPtr != None:

sys.stdout.write(indent)

if last:

sys.stdout.write("R----")

indent += " "

else:

sys.stdout.write("L----")

indent += "| "

print(currPtr.key)

self.printHelper(currPtr.left, indent, False)

self.printHelper(currPtr.right, indent, True)

myTree = AVLTree()

root = None

nums = [33, 13, 52, 9, 21, 61, 8, 11]

for num in nums:

root = myTree.insert_node(root, num)

myTree.printHelper(root, "", True)

key = 13

root = myTree.delete_node(root, key)

print("After Deletion: ")

myTree.printHelper(root, "", True)</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs">// AVL tree implementation in Java

// Create node

class Node {

int item, height;

Node left, right;

Node(int d) {

item = d;

height = 1;

}

// Tree class

class AVLTree {

Node root;

int height(Node N) {

if (N == null)

return 0;

return N.height;

}

int max(int a, int b) {

return (a > b) ? a : b;

}

Node rightRotate(Node y) {

Node x = y.left;

Node T2 = x.right;

x.right = y;

y.left = T2;

y.height = max(height(y.left), height(y.right)) + 1;

x.height = max(height(x.left), height(x.right)) + 1;

return x;

}

Node leftRotate(Node x) {

Node y = x.right;

Node T2 = y.left;

y.left = x;

x.right = T2;

x.height = max(height(x.left), height(x.right)) + 1;

y.height = max(height(y.left), height(y.right)) + 1;

return y;

}

// Get balance factor of a node

int getBalanceFactor(Node N) {

if (N == null)

return 0;

return height(N.left) - height(N.right);

}

// Insert a node

Node insertNode(Node node, int item) {

// Find the position and insert the node

if (node == null)

return (new Node(item));

if (item < node.item)

node.left = insertNode(node.left, item);

else if (item > node.item)

node.right = insertNode(node.right, item);

else

return node;

// Update the balance factor of each node

// And, balance the tree

node.height = 1 + max(height(node.left), height(node.right));

int balanceFactor = getBalanceFactor(node);

if (balanceFactor > 1) {

if (item < node.left.item) {

return rightRotate(node);

} else if (item > node.left.item) {

node.left = leftRotate(node.left);

return rightRotate(node);

}

if (balanceFactor < -1) {

if (item > node.right.item) {

return leftRotate(node);

} else if (item < node.right.item) {

node.right = rightRotate(node.right);

return leftRotate(node);

}

return node;

}

Node nodeWithMimumValue(Node node) {

Node current = node;

while (current.left != null)

current = current.left;

return current;

}

// Delete a node

Node deleteNode(Node root, int item) {

// Find the node to be deleted and remove it

if (root == null)

return root;

if (item < root.item)

root.left = deleteNode(root.left, item);

else if (item > root.item)

root.right = deleteNode(root.right, item);

else {

if ((root.left == null) || (root.right == null)) {

Node temp = null;

if (temp == root.left)

temp = root.right;

else

temp = root.left;

if (temp == null) {

temp = root;

root = null;

} else

root = temp;

} else {

Node temp = nodeWithMimumValue(root.right);

root.item = temp.item;

root.right = deleteNode(root.right, temp.item);

}

if (root == null)

return root;

// Update the balance factor of each node and balance the tree

root.height = max(height(root.left), height(root.right)) + 1;

int balanceFactor = getBalanceFactor(root);

if (balanceFactor > 1) {

if (getBalanceFactor(root.left) >= 0) {

return rightRotate(root);

} else {

root.left = leftRotate(root.left);

return rightRotate(root);

}

if (balanceFactor < -1) {

if (getBalanceFactor(root.right) <= 0) {

return leftRotate(root);

} else {

root.right = rightRotate(root.right);

return leftRotate(root);

}

return root;

}

void preOrder(Node node) {

if (node != null) {

System.out.print(node.item + " ");

preOrder(node.left);

preOrder(node.right);

}

// Print the tree

private void printTree(Node currPtr, String indent, boolean last) {

if (currPtr != null) {

System.out.print(indent);

if (last) {

System.out.print("R----");

indent += " ";

} else {

System.out.print("L----");

indent += "| ";

}

System.out.println(currPtr.item);

printTree(currPtr.left, indent, false);

printTree(currPtr.right, indent, true);

}

// Driver code

public static void main(String[] args) {

AVLTree tree = new AVLTree();

tree.root = tree.insertNode(tree.root, 33);

tree.root = tree.insertNode(tree.root, 13);

tree.root = tree.insertNode(tree.root, 53);

tree.root = tree.insertNode(tree.root, 9);

tree.root = tree.insertNode(tree.root, 21);

tree.root = tree.insertNode(tree.root, 61);

tree.root = tree.insertNode(tree.root, 8);

tree.root = tree.insertNode(tree.root, 11);

tree.printTree(tree.root, "", true);

tree.root = tree.deleteNode(tree.root, 13);

System.out.println("After Deletion: ");

tree.printTree(tree.root, "", true);

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp">// AVL tree implementation in C

#include <stdio.h>

#include <stdlib.h>

// Create Node

struct Node {

int key;

struct Node *left;

struct Node *right;

int height;

};

// Calculate height

int height(struct Node *N) {

if (N == NULL)

return 0;

return N->height;

}

return (a > b) ? a : b;

}

// Create a node

struct Node *newNode(int key) {

struct Node *node = (struct Node *)

malloc(sizeof(struct Node));

node->key = key;

node->left = NULL;

node->right = NULL;

node->height = 1;

return (node);

}

// Right rotate

struct Node *rightRotate(struct Node *y) {

struct Node *x = y->left;

struct Node *T2 = x->right;

x->right = y;

y->left = T2;

y->height = max(height(y->left), height(y->right)) + 1;

x->height = max(height(x->left), height(x->right)) + 1;

return x;

}

// Left rotate

struct Node *leftRotate(struct Node *x) {

struct Node *y = x->right;

struct Node *T2 = y->left;

y->left = x;

x->right = T2;

x->height = max(height(x->left), height(x->right)) + 1;

y->height = max(height(y->left), height(y->right)) + 1;

return y;

}

// Get the balance factor

int getBalance(struct Node *N) {

if (N == NULL)

return 0;

return height(N->left) - height(N->right);

}

// Insert node

struct Node *insertNode(struct Node *node, int key) {

// Find the correct position to insertNode the node and insertNode it

if (node == NULL)

return (newNode(key));

if (key < node->key)

node->left = insertNode(node->left, key);

else if (key > node->key)

node->right = insertNode(node->right, key);

else

return node;

// Update the balance factor of each node and

// Balance the tree

node->height = 1 + max(height(node->left),

height(node->right));

int balance = getBalance(node);

if (balance > 1 && key < node->left->key)

return rightRotate(node);

if (balance < -1 && key > node->right->key)

return leftRotate(node);

if (balance > 1 && key > node->left->key) {

node->left = leftRotate(node->left);

return rightRotate(node);

}

if (balance < -1 && key < node->right->key) {

node->right = rightRotate(node->right);

return leftRotate(node);

}

return node;

}

struct Node *minValueNode(struct Node *node) {

struct Node *current = node;

while (current->left != NULL)

current = current->left;

return current;

}

// Delete a nodes

struct Node *deleteNode(struct Node *root, int key) {

// Find the node and delete it

if (root == NULL)

return root;

if (key < root->key)

root->left = deleteNode(root->left, key);

else if (key > root->key)

root->right = deleteNode(root->right, key);

else {

if ((root->left == NULL) || (root->right == NULL)) {

struct Node *temp = root->left ? root->left : root->right;

if (temp == NULL) {

temp = root;

root = NULL;

} else

*root = *temp;

free(temp);

} else {

struct Node *temp = minValueNode(root->right);

root->key = temp->key;

root->right = deleteNode(root->right, temp->key);

}

if (root == NULL)

return root;

// Update the balance factor of each node and

// balance the tree

root->height = 1 + max(height(root->left),

height(root->right));

int balance = getBalance(root);

if (balance > 1 && getBalance(root->left) >= 0)

return rightRotate(root);

if (balance > 1 && getBalance(root->left) < 0) {

root->left = leftRotate(root->left);

return rightRotate(root);

}

if (balance < -1 && getBalance(root->right) <= 0)

return leftRotate(root);

if (balance < -1 && getBalance(root->right) > 0) {

root->right = rightRotate(root->right);

return leftRotate(root);

}

return root;

}

// Print the tree

void printPreOrder(struct Node *root) {

if (root != NULL) {

printf("%d ", root->key);

printPreOrder(root->left);

printPreOrder(root->right);

}

int main() {

struct Node *root = NULL;

root = insertNode(root, 2);

root = insertNode(root, 1);

root = insertNode(root, 7);

root = insertNode(root, 4);

root = insertNode(root, 5);

root = insertNode(root, 3);

root = insertNode(root, 8);

printPreOrder(root);

root = deleteNode(root, 3);

printf("\nAfter deletion: ");

printPreOrder(root);

return 0;

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs">// AVL tree implementation in C++

#include <iostream>

using namespace std;

class Node {

public:

int key;

Node *left;

Node *right;

int height;

};

// Calculate height

int height(Node *N) {

if (N == NULL)

return 0;

return N->height;

}

return (a > b) ? a : b;

}

// New node creation

Node *newNode(int key) {

Node *node = new Node();

node->key = key;

node->left = NULL;

node->right = NULL;

node->height = 1;

return (node);

}

// Rotate right

Node *rightRotate(Node *y) {

Node *x = y->left;

Node *T2 = x->right;

x->right = y;

y->left = T2;

y->height = max(height(y->left),

height(y->right)) +

1;

x->height = max(height(x->left),

height(x->right)) +

1;

return x;

}

// Rotate left

Node *leftRotate(Node *x) {

Node *y = x->right;

Node *T2 = y->left;

y->left = x;

x->right = T2;

x->height = max(height(x->left),

height(x->right)) +

1;

y->height = max(height(y->left),

height(y->right)) +

1;

return y;

}

// Get the balance factor of each node

int getBalanceFactor(Node *N) {

if (N == NULL)

return 0;

return height(N->left) -

height(N->right);

}

// Insert a node

Node *insertNode(Node *node, int key) {

// Find the correct postion and insert the node

if (node == NULL)

return (newNode(key));

if (key < node->key)

node->left = insertNode(node->left, key);

else if (key > node->key)

node->right = insertNode(node->right, key);

else

return node;

// Update the balance factor of each node and

// balance the tree

node->height = 1 + max(height(node->left),

height(node->right));

int balanceFactor = getBalanceFactor(node);

if (balanceFactor > 1) {

if (key < node->left->key) {

return rightRotate(node);

} else if (key > node->left->key) {

node->left = leftRotate(node->left);

return rightRotate(node);

}

if (balanceFactor < -1) {

if (key > node->right->key) {

return leftRotate(node);

} else if (key < node->right->key) {

node->right = rightRotate(node->right);

return leftRotate(node);

}

return node;

}

// Node with minimum value

Node *nodeWithMimumValue(Node *node) {

Node *current = node;

while (current->left != NULL)

current = current->left;

return current;

}

// Delete a node

Node *deleteNode(Node *root, int key) {

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