Number of Distinct Parenthesizations on Word

Theorem
Let $w$ be a word of $n$ elements.

The number of distinct parenthesizations of $w$ is the Catalan number $C_{n - 1}$:
 * $C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$