Example of proof by induction

Proof by strong induction on the height of the tree.

BASIS: n=0. There is only one tree of height 0. It has 1 node that is the root and also is a leaf.

L = number of leaves = 1, H = height = n = 0, L ≥ H+1 is true.

INDUCTION: Assume that the theorem is true for all binary trees of height ≤ n. We will prove that the theorem is true for all binary trees of height n+1.

Consider an arbitrary binary tree of height n+1 ≥ 1. This tree, T, has a root that is a non-leaf and hence has 2 sons (that are roots of binary trees).
Let the taller of these two subtrees be T₁ (it must have height n). The other subtree is T₂ and has height ≥ 0.

L₁ ≥ H₁+1 = n+1, by the inductive hypothesis.
L₂ ≥ H₂+1 ≥ 1, since H₂ ≥ 0.
L = L₁ + L₂ ≥ (n+1)+1 = n+2 ≥ (n+1) + 1.