Question

The number of trees in a binomial heap with \( n \) nodes is:

• A. \( \log n \)
• B. \( n \)
• C. \( n \log n \)
• D. \( n/2 \)

Hint:

The number of trees in a binomial heap with \( n \) nodes is at most \( O(\log n) \), as it corresponds to the number of set bits in the binary representation of \( n \).

Answer 1

The Correct Option is A

Step 1: Understanding a Binomial Heap
- A binomial heap is a collection of binomial trees that follow the min-heap or max-heap property.
- A binomial tree of order \( k \) has exactly \( 2^k \) nodes.
Step 2: Number of Trees in a Binomial Heap
- Any number \( n \) can be represented in binary form.
- The number of binomial trees in a binomial heap corresponds to the number of set bits (1s) in the binary representation of \( n \).
- The maximum number of binomial trees for \( n \) nodes is \( O(\log n) \).
Step 3: Evaluating the Options
- (A) Correct: The number of trees is at most \( \log n \).
- (B) Incorrect: The number of trees is much smaller than \( n \).
- (C) Incorrect: \( n \log n \) is incorrect as binomial heaps maintain logarithmic complexity.
- (D) Incorrect: \( n/2 \) is incorrect as the number of trees is based on binary representation.