A binary tree T has 64 leaf nodes. The number of nodes of degree 2 in T is —

The correct option is C (63).

Explanation

In any binary tree, there is a strict relationship between the number of leaf nodes (nodes with 0 children) and the number of nodes with degree 2 (nodes with 2 children).

Let:

• n₀ = number of leaf nodes (degree 0)

• n₁ = number of nodes with exactly 1 child (degree 1)

• n₂ = number of nodes with exactly 2 children (degree 2)

The total number of nodes (N) is:

N = n₀ + n₁ + n₂

In any tree, the total number of edges (E) is one less than the total number of nodes:

E = N - 1

Therefore,

E = n₀ + n₁ + n₂ - 1

We can also count edges based on the number of children:

• Each degree 0 node contributes 0 edges.

• Each degree 1 node contributes 1 edge.

• Each degree 2 node contributes 2 edges.

Hence,

E = n₁ + 2n₂

Equating both expressions for E:

n₀ + n₁ + n₂ - 1 = n₁ + 2n₂

Subtracting n₁ and n₂ from both sides:

n₀ - 1 = n₂

Conclusion

For any binary tree:

n₂ = n₀ - 1

That is, the number of nodes with two children is always exactly one less than the number of leaf nodes.

Given:
n₀ = 64

Therefore:

n₂ = 64 - 1 = 63

Hence, the correct answer is C (63).