Ears of triangulations and Catalan numbers

It is known that a convex polygon of n sides admits Cn-2 triangulations, where Cn is a Catalan number. We classify these triangulations (considered as outerplanar graphs) according to their dual trees, and prove the following formula for the number of triangulations of a convex n-gon whose dual tree has exactly k leaves: nk2n−2kn−42k−4Ck−2 The proof is bijective and provides a recursive formula for the Catalan numbers similar to, but different from, a classical identity of Touchard. An averaging argument allows one to deduce Touchardʹs formula from ours.