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

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NIELIT Scientific Assistant IT 5 Dec 2021 Official Paper
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  1. 2k − 1
  2. 2k+1 − 1 
  3. 2k−1 + 1
  4. 2k + 1

Answer (Detailed Solution Below)

Option 2 : 2k+1 − 1 
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The correct answer is option 2: 2k+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++2k=2k+11
  • This is the formula for the number of nodes in a perfect binary tree of height k.

Additional Information

  • 2k − 1: This would be valid if height was defined as number of levels minus one.
  • 2k−1 + 1 and 2k + 1: Incorrect and do not follow from the geometric progression formula.
  • Therefore, for height k starting from root at 0, maximum nodes = 2k+11

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

Latest NIELIT Scientific Assistant Updates

Last updated on Jul 3, 2025

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