A balanced binary tree with m elements will have height ______________ . A. 2m B. 2m C. logm 2 D. none of the above E. log2 m

To determine the height of a balanced binary tree with $m$ elements, we can break down the problem and analyze the concept of height in binary trees.

Step 1: Break Down the Problem

1. Identify the definition of the height of a binary tree.
2. Understand the characteristics of a balanced binary tree.
3. Relate the number of elements $m$ to the height $h$.

Step 2: Relevant Concepts

• The height $h$ of a balanced binary tree is the number of edges on the longest path from the root to a leaf.
• A balanced binary tree is structured such that all leaf nodes are at the same level, or their heights differ by one at most.

The relationship between height $h$ and the number of nodes $m$ in a complete binary tree can be expressed as: $m = 2^{h+1} - 1 \quad \text{(for a full binary tree)}$ From this relationship, we can derive that $h$ is related to $m$ as: $h \leq \log_2 (m + 1)$ This indicates the height will be approximately $\log_2 m$.

Step 3: Analysis and Detail

Since a balanced binary tree can essentially be filled perfectly (except possibly for the last level, which is filled from left to right):

• For $m$ nodes, the height would reach a maximum when the tree is highly balanced.
• The logarithmic relationship arises from the binary splitting of nodes, hence the expected height is logarithmic.

Given that the height $h$ is generally expressed in terms of the base 2 logarithm, we narrow down the possible options.

Step 4: Verify and Summarize

Options provided are:

• A. $2m$
• B. $2m$ (duplicate option)
• C. $\log_2 m$
• D. none of the above
• E. $\log_2 m$

Both option C and E are identical, thus we can confirm that $\log_2 m$ is the correct representation for the height of a balanced binary tree with $m$ elements.

Final Answer

The height of a balanced binary tree with $m$ elements is approximately: $\log_2 m$ Thus, the correct choice is: E. $\log_2 m$.