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.