Suppose a binary tree is constructed with n nodes, such that each node has exactly either zero or two children. The maximum height of the tree will be

To find the maximum height of a binary tree constructed with $n$ nodes where each node has either zero or two children, we can analyze the structure of such a tree. Below are the details:

1. ### Break Down the Problem

• We need to examine the characteristics of a binary tree with the specified conditions.
• Understand the relationship between the number of nodes and the height of the tree.

2. ### Relevant Concepts

• A binary tree's height is defined as the number of edges on the longest path from the root node down to the farthest leaf node.
• For a tree with $n$ nodes where every node has either 0 or 2 children, it is a full binary tree.
• In a full binary tree, each level is completely filled except possibly the last level.

3. ### Analysis and Detail

• In a full binary tree with $n$ nodes, the maximum number of nodes that can occur at each level $h$ of the tree is given by $2^h$.

• The total number of nodes in a complete binary tree of height $h$ can be calculated as:

  $\text{Total nodes} = 1 + 2 + 4 + \ldots + 2^h = 2^{h+1} - 1$

  where $h$ is the height of the tree.

• Rearranging the formula gives:

  $n = 2^{h+1} - 1 \implies 2^{h+1} = n + 1 \implies h + 1 = \log_2(n + 1) \implies h = \log_2(n + 1) - 1$

4. ### Verify and Summarize

• The maximum height of such a binary tree can be derived as $h = \log_2(n + 1) - 1$.
• This means that for any $n$ nodes, the formula accurately represents the height of the tree based on the conditions given.

Final Answer

The maximum height of a binary tree constructed with $n$ nodes, where each node has either zero or two children, is given by:

$h = \log_2(n + 1) - 1$