Some more properties of Catalan numbers

We want to illustrate some correspondences between Catalan numbers and combinatoric objects, such as plane walks, binary trees and some particular words. By means of under-diagonal walks, we give a combinatorial interpretation of the formula Cn = 1 n+1 2n n defining Catalan numbers. These numbers also enumerate both words in a particular language defined on a four character alphabet and the corresponding walks made up of four different types of steps. We illustrate a bijection between n-long words in this language and binary trees having n + 1 nodes, after which we give a simple proof of Touchard's formula.