GATE CS 2015 Set 2 Q.20 (CS Q.10): Binary Trees, Binary Search Trees and Balanced Trees

A binary tree $T$ has 20 leaves. The number of nodes in $T$ having two children is _____.

Your answer

Official answer and solution

Official answer: 19
From: Official GATE 2015 answer key

Solution

Apply the structural relation between leaves and degree-2 internal nodes in a binary tree.

Let $n$ be the total number of nodes in binary tree $T$ , and let $n_0, n_1, n_2$ denote the number of nodes with 0, 1, and 2 children, respectively.

The total number of nodes is:

n = n_0 + n_1 + n_2.

Every node except the root has exactly one incoming edge, so the total number of edges is $n - 1$ . The number of edges can also be counted from outgoing degrees:

n - 1 = 0 \cdot n_0 + 1 \cdot n_1 + 2 \cdot n_2 = n_1 + 2n_2.

Subtracting the two equations:

(n_0 + n_1 + n_2) - 1 = n_1 + 2n_2 \implies n_0 - 1 = n_2 \implies n_2 = n_0 - 1.

Given $n_0 = 20$ leaves:

n_2 = 20 - 1 = 19.

Final answer

Common trap

Confusing the relation for general binary trees ( $n_0 = n_2 + 1$ ) with that for strictly full binary trees.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 2
Question: 20 of 65, printed as CS Q.10
Section: Computer Science and Information Technology

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