Question

The height of any rooted tree is defined as the maximum number of edges in the path from the root node to any leaf node. Suppose a Min-Heap \( T \) stores 32 keys. The height of \( T \) is _________ (Answer in integer).

Hint:

In a complete binary tree, the height is the number of edges from the root to the deepest leaf. The height is approximately \( \log_2 n \), where \( n \) is the number of nodes.

Answer 1

Correct Answer: 5

A Min-Heap is a complete binary tree where each level is filled completely, except possibly the last level, which is filled from left to right. The height of a binary tree is the number of edges on the longest path from the root to any leaf node. To calculate the height of the tree, we use the following formula:
For a complete binary tree with \( n \) nodes, the height \( h \) is the greatest integer such that \( 2^h \leq n \). This is because the height of a binary tree is logarithmic with respect to the number of nodes. Step 1: Calculate the height for 32 keys
The total number of nodes in the tree is 32, and we need to find the height. Since the tree is a complete binary tree, the number of nodes at height \( h \) is \( 2^h \). To find the height \( h \), we calculate the greatest integer \( h \) such that: \[ 2^h \leq 32 \] We know: \[ 2^5 = 32 \] Thus, the height of the tree is 5. Step 2: Confirm the number of edges
For a tree with height 5, the maximum number of edges from the root to any leaf node is 5, as there are 5 levels from the root to the deepest leaf. Thus, the height of the Min-Heap storing 32 keys is 5.