Relationship between number of nodes and height of binary tree

Binary Tree is a hierarchical data structure in which each node has at most two children, referred to as the left child and the right child. It is commonly used in computer science for efficient storage and retrieval of data, with various operations such as insertion, deletion, and traversal.

Prerequisite - Binary Tree Data Structure

Height of Binary Tree

The height of the binary tree is the longest path from root node to any leaf node in the tree. For example, the height of binary tree shown in Figure 1(b) is 2 as longest path from root node to node 2 is 2. Also, the height of binary tree shown in Figure 1(a) is 4. 

Calculating minimum and maximum height from the number of nodes -

If there are n nodes in a binary tree, the maximum height of the binary tree is n-1, and the minimum height is floor (log2(n)). 
For example, the left skewed binary tree shown in Figure 1(a) with 5 nodes has a height of 5-1 = 4, and the binary tree shown in Figure 1(b) with 5 nodes has a height floor(log₂5) = 2. 

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Calculating minimum and maximum number of nodes from height:

If binary tree has height h, minimum number of nodes is h+1 (in case of left skewed and right skewed binary tree). For example, the binary tree shown in Figure 2(a) with height 2 has 3 nodes. 

If binary tree has height h, maximum number of nodes will be when all levels are completely full. Total number of nodes will be 2^0 + 2^1 + …. 2^h = 2^(h+1)-1. For example, the binary tree shown in Figure 2(b) with height 2 has 2^(2+1)-1 = 7 nodes.

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Questions on Finding Height of Binary Tree

Question-1.The height of a tree is the length of the longest root-to-leaf path in it. The maximum and the minimum number of nodes in a binary tree of height 5 are: 

• (A) 63 and 6, respectively 
• (B) 64 and 5, respectively 
• (C) 32 and 6, respectively 
• (D) 31 and 5, respectively 

Solution:

According to formula discussed, 
max number of nodes = 2^(h+1)-1 = 2^6-1 =63. 
min number of nodes = h+1 = 5+1 = 6. 

Question-2.Which of the following height is not possible for a binary tree with 50 nodes? 

• (A) 4 
• (B) 5 
• (C) 6 
• (D) None 

Solution:

According to formula discussed, 
Minimum height with 50 nodes = floor(log2(50)) = 5. Therefore, height 4 is not possible.