Prove that the number of full nodes plus one is qual to the number of leaves in a nonempty binary tree
Well, an induction proof consists of a base case and an induction step. For trees that is usually structural induction, based on inductive ways trees are built.

The base case considers the smallest binary tree, which cannot be composed further. Show that one has the Property.

The induction step argues that the Property is true for any tree $T$ assuming the Property holds for its subtrees $T_{\textrm{left}}$ and $T_{\textrm{right}}$. That assumption is called the induction hypothesis.

There is another approach, induction based on the number of (full) nodes, called natural induction. Base case has a single full node. Induction step argues Property is true assuming Property holds for any tree having less full nodes.