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The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height h is:
a)2^h -1
b)2^(h-1) – 1
c)2^(h+1) -1
d)2^(h+1)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The height of a binary tree is the maximum number of edges in any root...
Maximum number of nodes will be there for a complete tree. Number of nodes in a complete tree of height h = 1 + 2 + 2^2 + 2*3 + …. 2^h = 2^(h+1) – 1
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Most Upvoted Answer
The height of a binary tree is the maximum number of edges in any root...
C)2h+1 -1

The correct answer is a) 2h-1.

Explanation:

To understand why this is the correct answer, let's consider a few examples.

Example 1: A binary tree of height 1

1
/ \
2 3

Here, the height of the tree is 1. The maximum number of nodes in any root to leaf path is 2 (root -> left child, or root -> right child). The tree has a total of 3 nodes. Let's see if the formula 2h-1 holds true:

2h-1 = 2(1)-1 = 1 (which is the correct number of nodes in this tree)

Example 2: A binary tree of height 2

1
/ \
2 3
/ \
4 5

Here, the height of the tree is 2. The maximum number of nodes in any root to leaf path is 3 (root -> left child -> left child, or root -> left child -> right child, or root -> right child). The tree has a total of 5 nodes. Let's see if the formula 2h-1 holds true:

2h-1 = 2(2)-1 = 3 (which is the correct number of nodes in this tree)

Example 3: A binary tree of height 3

1
/ \
2 3
/ \
4 5
/ \
6 7

Here, the height of the tree is 3. The maximum number of nodes in any root to leaf path is 4 (root -> left child -> left child -> left child, or root -> left child -> left child -> right child, or root -> left child -> right child -> left child, or root -> left child -> right child -> right child, or root -> right child -> left child, or root -> right child -> right child). The tree has a total of 7 nodes. Let's see if the formula 2h-1 holds true:

2h-1 = 2(3)-1 = 7 (which is the correct number of nodes in this tree)

From these examples, we can see that the formula 2h-1 gives us the correct maximum number of nodes in a binary tree of height h. Therefore, the correct answer is a) 2h-1.
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