Finite combinatorial objects often come with notions of size and containment. It is natural to ask whether they can be constructed step by step in such a way that each object contains the last and is uniformly random among objects of its size.

For example: starting with a triangle drawn on the plane, can you add line segments two at a time so that at every stage you have a uniformly random triangulation of a polygon?

We will describe such processes for this and some other structures related to Catalan numbers. (Joint work with DIMACS visitor Malwina Luczak, now at Cambridge.)