Let's analyze the problem step-by-step to understand how the answer **511** is obtained. --- ## Problem Restatement: - The array is a binary min-heap with **1023 elements**. - The question asks: **"The minimum number of comparisons required to find the maximum in the heap is ____."** - The given correct answer is **511**. --- ## Step-by-step reasoning: ### 1. **Understanding the structure of a binary min-heap** - For **1023 elements**, the heap is a complete binary tree. - The height \( h \) of the heap: \[ h = \lfloor \log_2 1023 \rfloor = 9 \] (since \(2^9 = 512\), and \(1023 \approx 2^{10} - 1\)) - **Levels:** - Level (root): 1 node - Level 1: 2 nodes - ... - Level 9: 512 nodes - Total nodes: 1023 ### 2. **Properties of a Min-Heap** - The minimum element is at the root. - The maximum element is at one of the leaf nodes, because in a min-heap, only the root is guaranteed to be the minimum; the maximum can be among the leaves. ### 3. **Finding the maximum in a min-heap** - To find the maximum, we need to compare all **leaf nodes**. - The number of leaves in a complete binary tree with \( n \) nodes: \[ \text{Number of leaves} = \lceil \frac{n+1}{2} \rceil \] - For \( n = 1023 \): \[ \frac{1023 + 1}{2} = \frac{1024}{2} = 512 \] - So, **there are 512 leaves**. ### 4. **Number of comparisons needed** - To find the maximum among 512 leaves, **we need 511 comparisons** (since comparing pairs reduces the set to the maximum in \( N - 1 \) comparisons). - **Example:** - Comparing two leaves: 1 comparison - Comparing the result with the next leaf: 1 comparison - Continue until all leaves are processed: \[ \text{Total comparisons} = 512 - 1 = 511 \] ### 5. **Why is the answer 511?** - Because **comparing all the leaves** to find the maximum requires **512 - 1 = 511 comparisons**. --- ## **Final Answer:** \[ \boxed{511} \] This matches the provided correct answer. --- ## **Summary:** - The maximum element in a min-heap is among its leaves. - There are 512 leaves. - To find the maximum among these leaves, 511 comparisons are needed. --- Let me know if you'd like further clarification!