Balanced Binary Tree

Last Updated : 14 Aug, 2024

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A binary tree is balanced if the height of the tree is O(Log n) where n is the number of nodes. For Example, the AVL tree maintains O(Log n) height by making sure that the difference between the heights of the left and right subtrees is at most 1. Red-Black trees maintain O(Log n) height by making sure that the number of Black nodes on every root-to-leaf path is the same and that there are no adjacent red nodes. Balanced Binary Search trees are performance-wise good as they provide O(log n) time for search, insert and delete.

A single node is always balanced. It is also referred to as a height-balanced binary tree.
An empty tree (Root = Null) is also always considered as balanced.

Example:

balance-vs-unbalance-binnary-tree

It is a type of binary tree in which the difference between the height of the left and the right subtree for each node is either 0 or 1. In the figure above, the root node having a value 0 is unbalanced with a depth of 2 units.

How to Check if a Binary Tree is balanced?

To check if a Binary tree is balanced we need to check three conditions :

The absolute difference between heights of left and right subtrees at any node should be less than 1.
For each node, its left subtree should be a balanced binary tree.
For each node, its right subtree should be a balanced binary tree.

We can solve this problem in O(n) time. Please refer Balanced Binary Tree or Not for details.

Self-Balancing Binary Search Trees

In data structure and programming, we mainly discuss two self-balancing binary search trees, which are as follows:

AVL Trees

AVL tree is a self-balancing Binary Search Tree (BST) where the difference between heights of left and right subtrees cannot be more than one for all nodes.

Example of AVL Trees:

Example-of-AVL-Tree

The above tree is AVL because the differences between the heights of left and right subtrees for every node are less than or equal to 1.

Red Black Tree

A Red-Black Tree is a self-balancing binary search tree where each node has an additional attribute: a color, which can be either red or black. The primary objective of these trees is to maintain balance during insertions and deletions, ensuring efficient data retrieval and manipulation.

Example-of-Red-Black-Tree

The Red-Black Tree in above image ensures that every path from the root to a leaf node has the same number of black nodes. In this case, there is one (excluding the root node).

Advantages of Balanced Binary Tree:

Balanced binary trees, such as AVL trees and Red-Black trees, maintain their height in logarithmic proportion to the number of nodes. This ensures that fundamental operations like insertion, deletion, and search are executed with O(log n) time complexity.
Non Destructive In a balanced binary tree are performed in such a way that the tree remains balanced without requiring a complete reorganization.
Balanced binary trees are well-suited for range queries, where you need to find all elements within a specified range.

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Introduction to Height Balanced Binary Tree

Balanced Binary Tree or Not

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Consider the following AVL tree. 60 / \ 20 100 / \ 80 120 Which of the following is updated AVL tree after insertion of 70 A 70 / \ 60 100 / / \ 20 80 120 B 100 / \ 60 120 / \ / 20 70 80 C 80 / \ 60 100 / \ \ 20 70 120 D 80 / \ 60 100 / / \ 20 70 120 (A) A (B) B (C) C (D) D Answer: (C) Explanation: Refer following for steps used in AVL insertion. A

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