Combinatorial use of the Catalan numbers

The Catalan numbers count a number of structures that appear in mathematics like ordered trees, monotone paths, non-crossing partitions, unlabeled semiorders, etc. But one of the most interesting things about Catalan numbers is that they count the number of expressions that can be built from purely parenthesis like these.

()
(())
(() ()), ((()))
(() () ()), (() (())), ((()) ()), ((() ())), (((())))

In other words, these Catalan numbers can be used to count the number of Lisp expressions built purely from the parenthesis. For those of us interested in list processing, it is worth considering the Catalan numbers and the structures related to them.