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

Understanding Min-Heap and Tree Height

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 defined as the number of edges on the longest path from the root to any leaf node.

Formula for Height Calculation
For a complete binary tree with \( n \) nodes, the height \( h \) is given by the greatest integer such that:
\[ 2^h \leq n \] Since the height of a binary tree is logarithmic with respect to the number of nodes, we use the logarithm base 2.

Step 1: Calculate the Height for 32 Nodes
Given that the tree has 32 nodes, we determine \( h \) such that:
\[ 2^h \leq 32 \] We know that:
\[ 2^5 = 32 \] Hence, the height of the tree is 5.

Step 2: Confirm the Number of Edges
For a tree of height 5, the maximum number of edges from the root to any leaf node is also 5, as there are 5 levels from the root to the deepest leaf.

Final Answer: The height of the Min-Heap storing 32 keys is 5.