Balanced Binary Tree | DSA in C++ - Software Development PDF Download

Introduction

In the field of data structures and algorithms, balanced binary trees play a crucial role in efficient data storage and retrieval. In this article, we will explore the concept of balanced binary trees, discuss their advantages, and provide code examples in C++ to help beginners grasp the fundamental concepts.

What is a Balanced Binary Tree?

A balanced binary tree, also known as an AVL tree, is a type of binary search tree (BST) with the additional property that the heights of the left and right subtrees of any node differ by at most one. This property ensures that the tree remains balanced and provides efficient time complexity for various operations such as searching, inserting, and deleting elements.

Characteristics of a Balanced Binary Tree

The height of a balanced binary tree is approximately log(n), where n is the number of nodes in the tree.

The height difference between the left and right subtrees of any node is at most 1.

The left subtree of a node contains values smaller than the node, while the right subtree contains values greater than the node.

The balanced property guarantees efficient search, insertion, and deletion operations with time complexity of O(log(n)).

Examples of Balanced Binary Trees

Here are a few examples of balanced binary trees:

Example 1:

     5

   /   \

  3     7

 / \   / \

2   4 6   8

Example 2:

      8

     / \

    6   9

   / \

  3   7

     / \

    5   10

Implementing a Balanced Binary Tree in C++

To implement a balanced binary tree, we can create a class that represents a node in the tree. Each node contains a value, pointers to its left and right child nodes, and a height attribute.

Example code:

class Node {

public:

    int value;

    Node* left;

    Node* right;

    int height;

    Node(int val) {

        value = val;

        left = right = nullptr;

        height = 1;

    }

};

Operations on Balanced Binary Trees

• Insertion: To insert a new element into a balanced binary tree, we follow the standard BST insertion process while ensuring the balance property is maintained. If the balance factor of any node becomes greater than 1 or less than -1, we perform rotations to restore balance.
• Deletion: Similar to insertion, deletion in a balanced binary tree involves maintaining the balance property. After deleting a node, we perform rotations if necessary to restore balance.
• Searching: Searching in a balanced binary tree follows the standard BST search algorithm. We compare the target value with the node's value and traverse left or right accordingly until we find a match or reach a leaf node.

Sample Problems and Solutions

Problem 1: Determine if a given binary tree is balanced.

bool isBalanced(Node* root) {

    if (root == nullptr)

        return true;

    int balanceFactor = getHeight(root->left) - getHeight(root->right);

    return (abs(balanceFactor) <= 1) && isBalanced(root->left) && isBalanced(root->right);

}

Problem 2: Find the height of a balanced binary tree.

int getHeight(Node* root) {

    if (root == nullptr)

        return 0;

    return max(getHeight(root->left), getHeight(root->right)) + 1;

}

Conclusion

Balanced binary trees are an important data structure in computer science, providing efficient search, insertion, and deletion operations. By maintaining balance, these trees ensure optimal performance for a wide range of applications. Understanding their implementation and characteristics is crucial for mastering data structures and algorithms.