If one takes a fully parenthesized sum of terms and removes actual terms and plus signs, then one gets a well-matched string of pairs of ('s and )'s. Moreover, different parethetizations yield different strings and all strings can be obtained in this way. Thus, ( is the th Catalan number).

To show the first fact, assume that is a fully parenthesized sum of terms and proceed by (strong) induction on starting with .