Question

In a binary tree, what is the maximum number of nodes at level k?

• A. $k^2$
• B. $2^k$
• C. $2k$
• D. $k!$

Hint:

The number of nodes at level \( k \) in a binary tree is given by \( 2^k \), as each level doubles the number of nodes from the previous level.

Answer 1

The Correct Option is B

In a binary tree, the maximum number of nodes at level \( k \) is \( 2^k \). Each level in a binary tree can have at most double the number of nodes as the previous level, starting with 1 node at level 0 (the root). For example, level 1 has 2 nodes, level 2 has 4 nodes, and so on.
- $k^2$ (A) is incorrect because the number of nodes does not grow quadratically with level number in a binary tree.
- $2k$ (C) is also incorrect, as the number of nodes grows exponentially, not linearly with \( k \).
- $k!$ (D) is incorrect because the factorial function does not describe the number of nodes at each level of a binary tree.
Thus, the correct answer is (B), \( 2^k \).