number of triangulations of a convex (n+2)-gon is catalan

Statement

If we have a convex polygon with $n + 2$ vertices, then a triangulation can be created by drawing diagonals inside the polygon to create triangles within. Note that lines cannot cross each other inside this polygon. Note also that we fix the vertices, so rotations are counted as distinct triangulations.

Then, the number of triangulation of a convex $(n + 2)$ -gon is equal to $C_{n}$ (the Catalan numbers).

Proof

We shall construct a binary tree from a triangulation and use the fact that the number of complete binary trees with n+1 leaves is catalan.

Given a triangulation of a convex $(n + 2)$ -gon:

Draw one vertex within each triangle
For all except for one side of the polygon, draw a vertex outside of it.
Connect vertices if they share a side between them