Maximum number of nodes in a binary tree with height k, where root is height 0, is :

Question

Question

This question was previously asked in

NIELIT Scientific Assistant IT 5 Dec 2021 Official Paper

Answer 1

The correct answer is option 2: 2^k+1 − 1

Key Points

In a binary tree, each level can contain at most twice the number of nodes as the previous level.
If height k means the root is at level 0 and the last level is k, then total levels = k + 1.
The maximum number of nodes in such a binary tree is given by the sum of the geometric progression:
- Nodes = $1 + 2 + 4 + \dots + 2^k = 2^{k+1} - 1$
This is the formula for the number of nodes in a perfect binary tree of height k.

Additional Information

2^k − 1: This would be valid if height was defined as number of levels minus one.
2^k−1 + 1 and 2^k + 1: Incorrect and do not follow from the geometric progression formula.
Therefore, for height k starting from root at 0, maximum nodes = $2^{k+1} - 1$

Hence, the correct answer is: option 2: 2^k+1 − 1