Height of Node MCQ Quiz - Objective Question with Answer for Height of Node - Download Free PDF

Last updated on Jun 12, 2025

Latest Height of Node MCQ Objective Questions

Height of Node Question 1:

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

  1. 2k − 1
  2. 2k+1 − 1 
  3. 2k−1 + 1
  4. 2k + 1

Answer (Detailed Solution Below)

Option 2 : 2k+1 − 1 

Height of Node Question 1 Detailed Solution

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" id="MathJax-Element-203-Frame" role="presentation" style="position: relative;" tabindex="0">1+2+4++2k=2k+111+2+4+⋯+2k=2k+1−1
  • 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" id="MathJax-Element-204-Frame" role="presentation" style="position: relative;" tabindex="0">2k+112k+1−1

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

Height of Node Question 2:

Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of T are:

Note: The height of a tree with a single node is 0.

  1. 4 and 15 respectively
  2. 3 and 14 respectively
  3. 4 and 14 respectively
  4. 3 and 15 respectively

Answer (Detailed Solution Below)

Option 2 : 3 and 14 respectively

Height of Node Question 2 Detailed Solution

Concepts:

Minimum height of the tree is when all the levels of BST are completely filled.

Maximum height of the binary search tree (BST) is the worst case when nodes are in skewed manner.

Formula:

Minimum height of the BST with n nodes is ⌈log2 (n + 1)⌉ - 1

Maximum height of the BST with n nodes is n - 1.

Calculation:

Maximum height of the BST with 15 nodes 15 - 1 = 14

Diagram:

Minimum height of the BST with n nodes is ⌈log2 (15 + 1)⌉ - 1 = 3

Diagram: 

Height of Node Question 3:

If root node is at the height 0 and the height of a binary tree T is 5 then what is the maximum number of nodes in a binary tree?

Answer (Detailed Solution Below) 63

Height of Node Question 3 Detailed Solution

Graph with height 5:

The maximum number of nodes present: 63 

Tips and Tricks:

If n is number of nodes and h is minimum height of in a binary search tree, then

n = 2h+1 – 1

n = 25+1 – 1

∴ n = 63

Top Height of Node MCQ Objective Questions

Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of T are:

Note: The height of a tree with a single node is 0.

  1. 4 and 15 respectively
  2. 3 and 14 respectively
  3. 4 and 14 respectively
  4. 3 and 15 respectively

Answer (Detailed Solution Below)

Option 2 : 3 and 14 respectively

Height of Node Question 4 Detailed Solution

Download Solution PDF

Concepts:

Minimum height of the tree is when all the levels of BST are completely filled.

Maximum height of the binary search tree (BST) is the worst case when nodes are in skewed manner.

Formula:

Minimum height of the BST with n nodes is ⌈log2 (n + 1)⌉ - 1

Maximum height of the BST with n nodes is n - 1.

Calculation:

Maximum height of the BST with 15 nodes 15 - 1 = 14

Diagram:

Minimum height of the BST with n nodes is ⌈log2 (15 + 1)⌉ - 1 = 3

Diagram: 

Height of Node Question 5:

Let T be a binary search tree with 15 nodes. The minimum and maximum possible heights of T are:

Note: The height of a tree with a single node is 0.

  1. 4 and 15 respectively
  2. 3 and 14 respectively
  3. 4 and 14 respectively
  4. 3 and 15 respectively

Answer (Detailed Solution Below)

Option 2 : 3 and 14 respectively

Height of Node Question 5 Detailed Solution

Concepts:

Minimum height of the tree is when all the levels of BST are completely filled.

Maximum height of the binary search tree (BST) is the worst case when nodes are in skewed manner.

Formula:

Minimum height of the BST with n nodes is ⌈log2 (n + 1)⌉ - 1

Maximum height of the BST with n nodes is n - 1.

Calculation:

Maximum height of the BST with 15 nodes 15 - 1 = 14

Diagram:

Minimum height of the BST with n nodes is ⌈log2 (15 + 1)⌉ - 1 = 3

Diagram: 

Height of Node Question 6:

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

  1. 2k − 1
  2. 2k+1 − 1 
  3. 2k−1 + 1
  4. 2k + 1

Answer (Detailed Solution Below)

Option 2 : 2k+1 − 1 

Height of Node Question 6 Detailed Solution

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" id="MathJax-Element-203-Frame" role="presentation" style="position: relative;" tabindex="0">1+2+4++2k=2k+111+2+4+⋯+2k=2k+1−1
  • 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" id="MathJax-Element-204-Frame" role="presentation" style="position: relative;" tabindex="0">2k+112k+1−1

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

Height of Node Question 7:

If root node is at the height 0 and the height of a binary tree T is 5 then what is the maximum number of nodes in a binary tree?

Answer (Detailed Solution Below) 63

Height of Node Question 7 Detailed Solution

Graph with height 5:

The maximum number of nodes present: 63 

Tips and Tricks:

If n is number of nodes and h is minimum height of in a binary search tree, then

n = 2h+1 – 1

n = 25+1 – 1

∴ n = 63

Height of Node Question 8:

How many nodes splitting operations will be performed if B, I, K are inserted into the following B tree of  order 4. Also find the no. of nodes in the resultant tree?

  1. 3 splits, 10 nodes
  2. 4 splits, 9 nodes
  3. 2 splits, 8 nodes
  4. 3 splits, 9 nodes

Answer (Detailed Solution Below)

Option 4 : 3 splits, 9 nodes

Height of Node Question 8 Detailed Solution

Insert B - 2 split required

Insert I - 1 split required

Insert K – 0 split required

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