Prove that in a tree where each node has 0 or 2 children, number of nodes with 0 child is one more than the number of nodes with 2 children
This can be proven by induction. What you describe is a Full binary tree, and it has a convenient induction definition:

a single node (or leaf) is a full binary tree;
a tree with two children that are full binary trees is a full binary tree.

This permits a structural induction proof of the property you want:

prove that it is true for a single node tree;
suppose that it is true for two full binary trees $a$, $b$, then prove that it is true for the node $N(a, b)$ with $a$ and $b$ as children.