Parenthesization/Examples/3

Example of Parenthesization
A word of $3$ elements can be parenthesized in $2$ distinct ways:
 * $\quad a_1 \left({a_2 a_3}\right)$
 * $\quad \left({a_1 a_2}\right) a_3$

Proof
From Number of Distinct Parenthesizations on Word, the number of distinct parenthesizations of a word $w$ of $n$ elements is the Catalan number $C_{n - 1}$:
 * $C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$

For $n = 3$ we have: