Question

Consider a binary tree in which every node has either 0 or 2 children. Let \( N>0 \) be the number of nodes in the tree. The number of nodes that have exactly 2 children is:

• A. \( \frac{N+1}{2} \)
• B. \( \frac{N-2}{2} \)
• C. \( \frac{N}{2} \)
• D. \( \frac{N-1}{2} \)

Hint:

In full binary trees (binary trees where every node has either 0 or 2 children), the number of nodes with 2 children is given by \( \frac{N-1}{2} \), where \( N \) is the total number of nodes in the tree.

Answer 1

The Correct Option is D

Step 1: Understanding the structure of the binary tree.

In a binary tree where each node has either 0 or 2 children, the total number of leaf nodes (nodes with 0 children) will be one more than the number of internal nodes (nodes with 2 children). This follows from the property of full binary trees.

Let:

- \( x \) be the number of internal nodes (with 2 children), - \( y \) be the number of leaf nodes (with 0 children).

For every binary tree: \[ x + y = N \quad \text{(total nodes)}. \] Since the number of leaf nodes is one more than the number of internal nodes: \[ y = x + 1. \] Thus, we have: \[ x + (x + 1) = N \quad \Rightarrow \quad 2x + 1 = N \quad \Rightarrow \quad x = \frac{N-1}{2}. \] Therefore, the number of nodes with exactly 2 children is \( \frac{N-1}{2} \). Thus, the correct answer is \( \frac{N-1}{2} \).