C++ Program to Implement AVL Tree
Last Updated :
23 Jul, 2025
AVL Tree, named after its inventors Adelson-Velsky and Landis, is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one, which ensures that the tree remains approximately balanced, providing efficient search, insertion, and deletion operations.
In this article, we will learn about the implementation of AVL Tree in C++, its basic operation and its applications.
What is an AVL Tree?
An AVL tree maintains balance by performing rotations during insertions and deletions to ensure the height difference between the left and right subtrees of any node is no more than one. This property guarantees that the tree's height remains O(logn), ensuring efficient operations.
An AVL Tree is a binary search tree with the following properties:
- The heights of the left and right subtrees of every node differ by at most one.
- Every subtree is an AVL tree.
- For every node, its balance factor (height of left subtree - height of right subtree) is -1, 0, or 1.
Implementation of AVL Tree in C++
An AVL tree can be implemented using a binary tree structure where each node will have left and right pointers and key values to store the data but along with that, we will store the height for each node so that the balance factor can be calculated easily. The balance factor of a node will be calculated for each node as the difference between the heights of its left and right subtrees.
When the balance factor for any node is not in the allowed limits, rotations are preformed to balance it.
AVL Tree Rotations
Rotations are the most important part of the working of the AVL tree. They are responsible for maintaining the balance in the AVL tree. There are 4 types of rotations based on the 4 possible cases:
- Right Rotation (RR)
- Left Rotation (LL)
- Left-Right Rotation (LR)
- Right-Left Rotation (RL)
Right Rotation (RR)
The Right Rotation (RR) is applied in an AVL tree when a node becomes unbalanced due to an insertion into the right subtree of its right child, leading to a Left Imbalance. To correct this imbalance, the unbalanced node is rotated 90° to the right (clockwise) along the top edge connected to its parent.
Left Rotation (LL)
The Left Rotation (LL) is used to balance a node that becomes unbalanced due to an insertion into the left subtree of its left child, also resulting in a Left Imbalance. The solution is to rotate the unbalanced node 90° to the left (anti-clockwise) along the top edge connected to its parent.
![left-rotation-ll-of-avl-tree[1]](https://media.geeksforgeeks.org/wp-content/uploads/20240715095423/left-rotation-ll-of-avl-tree%5B1%5D.webp)
Left-Right Rotation (LR)
The Left-Right Rotation (LR) is necessary when the left child of a node is right-heavy, creating a double imbalance. This situation is resolved by performing a left rotation on the left child, followed by a right rotation on the original node.
![left-right-rotation-in-avl-tree[1]](https://media.geeksforgeeks.org/wp-content/uploads/20240715095443/left-right-rotation-in-avl-tree%5B1%5D.webp)
Right-Left Rotation (RL)
The Right-Left Rotation (RL) is used when the right child of a node is left-heavy. This imbalance is corrected by performing a right rotation on the right child, followed by a left rotation on the original node.
![right-left-rotation-in-avl-tree[1]](https://media.geeksforgeeks.org/wp-content/uploads/20240715095500/right-left-rotation-in-avl-tree%5B1%5D.webp)
Representation of AVL Tree in C++
The following diagram represents the structure of an AVL Tree where the balance factor of each node is 0 and the tree is balanced:
To represent an AVL Tree in C++, we will use a class AVLNode to define the node structure and a class AVLTree to implement the AVL tree operations. We will use the templates in order to keep the AVL tree generic so that it can store multiple data types.
template <typename T>
class AVLNode {
public:
T key;
AVLNode* left;
AVLNode* right;
int height;
}
here:
- key: represents the value stored inside the node.
- left &right: are pointers to the left and right node.
- height: represents the height of each subtree starting from each node.
Implementation of Basic Operations of an AVL Tree in C++
Following are the basic operations that are required to work with an AVL tree:
Operation Name | Description | Time Complexity | Space Complexity |
|---|
Insert | Inserts a new element into the tree | O(log n) | O(log n) |
|---|
Delete Node | Removes an element from the tree | O(log n) | O(log n) |
|---|
Search | Searches for an element in the tree | O(log n) | O(1) |
|---|
Rotate Left | Performs left rotation to balance the AVL tree | O(1) | O(1) |
|---|
Rotate Right | Performs right rotation to balance the AVL tree | O(1) | O(1) |
|---|
Implementation of Insert Function
- Start at the root.
- Compare the new value with the current node.
- If less, move to the left child. If greater, move to the right child.
- Repeat until reaching a null position.
- Insert the new node at this position.
- Update the height of the current node.
- Calculate the balance factor of the current node.
- If the balance factor is >1 or <-1, perform necessary rotations:
- Left-Left case: Right rotation
- Left-Right case: Left rotation on left child, then right rotation.
- Right-Right case: Left rotation
- Right-Left case: Right rotation on right child, then left rotation.
- Repeat steps 6-8 while moving back up to the root.
Implementation of Delete Node Function
- Start at the root.
- Search for the node to delete.
- If the node is a leaf, simply remove it.
- If the node has one child, replace it with its child.
- If the node has two children:
- Find the in-order successor (minimum value in right subtree).
- Replace the node to be deleted with the in-order successor.
- Delete the in-order successor from its original position.
- Update the height of the current node.
- Calculate the balance factor of the current node.
- If the balance factor is >1 or <-1, perform necessary rotations:
- Left-Left case: Right rotation
- Left-Right case: Left rotation on left child, then right rotation
- Right-Right case: Left rotation
- Right-Left case: Right rotation on right child, then left rotation
- Repeat steps 6-8 while moving back up to the root.
Implementation of Search Function
- Start from the root.
- Compare the value with the current node.
- If equal, return true.
- If less, move to the left child.
- If greater, move to the right child.
- Repeat until found or reached a leaf node.
Implementation of Rotate Left Function
- Start with a node A that has a right child B.
- Make B's left subtree the right subtree of A.
- Make A the left child of B.
- Update the heights of A and B.
- Return B as the new root of this subtree.
Implementation of Rotate Right Function
- Start with a node A that has a left child B.
- Make B's right subtree the left subtree of A.
- Make A the right child of B.
- Update the heights of A and B.
- Return B as the new root of this subtree.
C++ Program to Implement AVL Tree
The following program demonstrates the implementation of AVL tree in C++:
C++
// C++ Program to Implement AVL Tree
#include <algorithm>
#include <iostream>
using namespace std;
// Template class representing a node in the AVL tree
template <typename T> class AVLNode {
public:
T key; // The value of the node
AVLNode* left; // Pointer to the left child
AVLNode* right; // Pointer to the right child
int height; // Height of the node in the tree
// Constructor to initialize a node with a given key
AVLNode(T k)
: key(k)
, left(nullptr)
, right(nullptr)
, height(1)
{
}
};
// Template class representing the AVL tree
template <typename T> class AVLTree {
private:
// Pointer to the root of the tree
AVLNode<T>* root;
// function to get the height of a node
int height(AVLNode<T>* node)
{
if (node == nullptr)
return 0;
return node->height;
}
// function to get the balance factor of a node
int balanceFactor(AVLNode<T>* node)
{
if (node == nullptr)
return 0;
return height(node->left) - height(node->right);
}
// function to perform a right rotation on a subtree
AVLNode<T>* rightRotate(AVLNode<T>* y)
{
AVLNode<T>* x = y->left;
AVLNode<T>* T2 = x->right;
// Perform rotation
x->right = y;
y->left = T2;
// Update heights
y->height
= max(height(y->left), height(y->right)) + 1;
x->height
= max(height(x->left), height(x->right)) + 1;
// Return new root
return x;
}
// function to perform a left rotation on a subtree
AVLNode<T>* leftRotate(AVLNode<T>* x)
{
AVLNode<T>* y = x->right;
AVLNode<T>* T2 = y->left;
y->left = x;
x->right = T2;
// Update heights
x->height
= max(height(x->left), height(x->right)) + 1;
y->height
= max(height(y->left), height(y->right)) + 1;
// Return new root
return y;
}
// function to insert a new key into the subtree rooted
// with node
AVLNode<T>* insert(AVLNode<T>* node, T key)
{
// Perform the normal BST insertion
if (node == nullptr)
return new AVLNode<T>(key);
if (key < node->key)
node->left = insert(node->left, key);
else if (key > node->key)
node->right = insert(node->right, key);
else
return node;
// Update height of this ancestor node
node->height = 1
+ max(height(node->left),
height(node->right));
// Get the balance factor of this ancestor node
int balance = balanceFactor(node);
// If this node becomes unbalanced, then there are 4
// cases
// Left Left Case
if (balance > 1 && key < node->left->key)
return rightRotate(node);
// Right Right Case
if (balance < -1 && key > node->right->key)
return leftRotate(node);
// Left Right Case
if (balance > 1 && key > node->left->key) {
node->left = leftRotate(node->left);
return rightRotate(node);
}
// Right Left Case
if (balance < -1 && key < node->right->key) {
node->right = rightRotate(node->right);
return leftRotate(node);
}
return node;
}
// function to find the node with the minimum key value
AVLNode<T>* minValueNode(AVLNode<T>* node)
{
AVLNode<T>* current = node;
while (current->left != nullptr)
current = current->left;
return current;
}
// function to delete a key from the subtree rooted with
// root
AVLNode<T>* deleteNode(AVLNode<T>* root, T key)
{
// Perform standard BST delete
if (root == nullptr)
return root;
if (key < root->key)
root->left = deleteNode(root->left, key);
else if (key > root->key)
root->right = deleteNode(root->right, key);
else {
// Node with only one child or no child
if ((root->left == nullptr)
|| (root->right == nullptr)) {
AVLNode<T>* temp
= root->left ? root->left : root->right;
if (temp == nullptr) {
temp = root;
root = nullptr;
}
else
*root = *temp;
delete temp;
}
else {
AVLNode<T>* temp
= minValueNode(root->right);
root->key = temp->key;
root->right
= deleteNode(root->right, temp->key);
}
}
if (root == nullptr)
return root;
// Update height of the current node
root->height = 1
+ max(height(root->left),
height(root->right));
// Get the balance factor of this node
int balance = balanceFactor(root);
// If this node becomes unbalanced, then there are 4
// cases
// Left Left Case
if (balance > 1 && balanceFactor(root->left) >= 0)
return rightRotate(root);
// Left Right Case
if (balance > 1 && balanceFactor(root->left) < 0) {
root->left = leftRotate(root->left);
return rightRotate(root);
}
// Right Right Case
if (balance < -1 && balanceFactor(root->right) <= 0)
return leftRotate(root);
// Right Left Case
if (balance < -1
&& balanceFactor(root->right) > 0) {
root->right = rightRotate(root->right);
return leftRotate(root);
}
return root;
}
// function to perform inorder traversal of the tree
void inorder(AVLNode<T>* root)
{
if (root != nullptr) {
inorder(root->left);
cout << root->key << " ";
inorder(root->right);
}
}
// function to search for a key in the subtree rooted
// with root
bool search(AVLNode<T>* root, T key)
{
if (root == nullptr)
return false;
if (root->key == key)
return true;
if (key < root->key)
return search(root->left, key);
return search(root->right, key);
}
public:
// Constructor to initialize the AVL tree
AVLTree()
: root(nullptr)
{
}
// Function to insert a key into the AVL tree
void insert(T key) { root = insert(root, key); }
// Function to remove a key from the AVL tree
void remove(T key) { root = deleteNode(root, key); }
// Function to search for a key in the AVL tree
bool search(T key) { return search(root, key); }
// Function to print the inorder traversal of the AVL
// tree
void printInorder()
{
inorder(root);
cout << endl;
}
};
int main()
{
AVLTree<int> avl;
// Insert nodes into the AVL tree
avl.insert(10);
avl.insert(20);
avl.insert(30);
avl.insert(40);
avl.insert(50);
avl.insert(25);
// Print the inorder traversal of the AVL tree
cout << "Inorder traversal of the AVL tree: ";
avl.printInorder();
// Remove a node from the AVL tree
avl.remove(30);
cout << "Inorder traversal after removing 30: ";
avl.printInorder();
// Search for nodes in the AVL tree
cout << "Is 25 in the tree? "
<< (avl.search(25) ? "Yes" : "No") << endl;
cout << "Is 30 in the tree? "
<< (avl.search(30) ? "Yes" : "No") << endl;
return 0;
}
OutputInorder traversal of the AVL tree: 10 20 25 30 40 50
Inorder traversal after removing 30: 10 20 25 40 50
Is 25 in the tree? Yes
Is 30 in the tree? No
Applications of AVL Tree
Following are some of the common applications of an AVL Tree:
- Database Indexing: AVL trees are used in database systems for efficient indexing and quick data retrieval.
- File Systems: Some file systems use AVL trees to maintain directory structures for fast file lookups.
- Spatial Indexing: In computer graphics and computational geometry, AVL trees are used for spatial indexing of objects.
- Network Routing Tables: AVL trees can be used to implement routing tables in network routers for efficient IP lookup.
- Priority Queues: While not as common as heaps, AVL trees can be used to implement priority queues with efficient operations.
- Symbol Tables in Compilers: Compilers often use AVL trees to implement symbol tables for fast insertion, deletion, and lookup of identifiers.
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Difference Between Call by Value and Call by Reference in CFunctions can be invoked in two ways: Call by Value or Call by Reference. These two ways are generally differentiated by the type of values passed to them as parameters.The following table lists the differences between the call-by-value and call-by-reference methods of parameter passing.Call By Valu
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Default Arguments in C++A default argument is a value provided for a parameter in a function declaration that is automatically assigned by the compiler if no value is provided for those parameters in function call. If the value is passed for it, the default value is overwritten by the passed value.Example:C++#include <i
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Inline Functions in C++In C++, inline functions provide a way to optimize the performance of the program by reducing the overhead related to a function call. When a function is specified as inline the whole code of the inline function is inserted or substituted at the point of its call during the compilation instead of us
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Lambda Expression in C++C++ 11 introduced lambda expressions to allow inline functions which can be used for short snippets of code that are not going to be reused. Therefore, they do not require a name. They are mostly used in STL algorithms as callback functions.Example:C++#include <iostream> using namespace std; i
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C++ Pointers and References
Pointers and References in C++In C++ pointers and references both are mechanisms used to deal with memory, memory address, and data in a program. Pointers are used to store the memory address of another variable whereas references are used to create an alias for an already existing variable. Pointers in C++ Pointers in C++ are a
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C++ PointersA pointer is a special variable that holds the memory address of another variable, rather than storing a direct value itself. Pointers allow programs to access and manipulate data in memory efficiently, making them a key feature for system-level programming and dynamic memory management. When we acc
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Dangling, Void , Null and Wild Pointers in CIn C programming pointers are used to manipulate memory addresses, to store the address of some variable or memory location. But certain situations and characteristics related to pointers become challenging in terms of memory safety and program behavior these include Dangling (when pointing to deall
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Applications of Pointers in CPointers in C are variables that are used to store the memory address of another variable. Pointers allow us to efficiently manage the memory and hence optimize our program. In this article, we will discuss some of the major applications of pointers in C. Prerequisite: Pointers in C. C Pointers Appl
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Understanding nullptr in C++Consider the following C++ program that shows problem with NULL (need of nullptr) CPP // C++ program to demonstrate problem with NULL #include <bits/stdc++.h> using namespace std; // function with integer argument void fun(int N) { cout << "fun(int)"; return;} // Overloaded fun
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References in C++In C++, a reference works as an alias for an existing variable, providing an alternative name for it and allowing you to work with the original data directly.Example:C++#include <iostream> using namespace std; int main() { int x = 10; // ref is a reference to x. int& ref = x; // printing v
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Can References Refer to Invalid Location in C++?Reference Variables: You can create a second name for a variable in C++, which you can use to read or edit the original data contained in that variable. While this may not sound appealing at first, declaring a reference and assigning it a variable allows you to treat the reference as if it were the
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Pointers vs References in C++Prerequisite: Pointers, References C and C++ support pointers, which is different from most other programming languages such as Java, Python, Ruby, Perl and PHP as they only support references. But interestingly, C++, along with pointers, also supports references. On the surface, both references and
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Passing By Pointer vs Passing By Reference in C++In C++, we can pass parameters to a function either by pointers or by reference. In both cases, we get the same result. So, what is the difference between Passing by Pointer and Passing by Reference in C++?Let's first understand what Passing by Pointer and Passing by Reference in C++ mean:Passing by
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When do we pass arguments by pointer?In C, the pass-by pointer method allows users to pass the address of an argument to the function instead of the actual value. This allows programmers to change the actual data from the function and also improve the performance of the program. In C, variables are passed by pointer in the following ca
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