In a binary tree where every node has zero or two children, if $n$ is the total number of nodes, how many nodes have exactly two children?

The correct option is (n-1)/2.

Let's analyze the structure of the binary tree step-by-step to understand why this formula holds true.
In a binary tree where every node has either zero or two children, the tree is known as a strict (or full) binary tree. Let:
- $n_0$ be the number of leaf nodes (nodes with 0 children).
- $n_2$ be the number of internal nodes with exactly 2 children.
- $n$ be the total number of nodes in the tree.

Since every node in this tree has either 0 or 2 children, there are no nodes with exactly 1 child. Therefore, the total number of nodes is the sum of the leaf nodes and the nodes with two children:
$n=n_0+n_2$

Now, let's look at the relationship between the number of edges and the nodes in any tree:
1. A tree with $n$ nodes has exactly $n-1$ edges.
2. We can also count the number of edges by looking at the outgoing branches from the nodes. A leaf node ($n_0$) contributes 0 outgoing edges, and a node with two children ($n_2$) contributes 2 outgoing edges. Therefore, the total number of edges is:
$\text{Total Edges}=2n_2$

Equating the two expressions for the total number of edges gives:
$n-1=2n_2$

To find the number of nodes with exactly two children ($n_2$), we solve the equation for $n_2$:
$n_2=\frac{n-1}{2}$

Thus, the number of nodes with exactly two children is indeed $(n-1)/2$.