Parenthesization/Examples/4

Example of Parenthesization
A word of $4$ elements can be parenthesized in $5$ distinct ways:
 * $\quad a_1 \paren {a_2 \paren {a_3 a_4} }$
 * $\quad a_1 \paren {\paren {a_2 a_3} a_4}$
 * $\quad \paren {a_1 a_2} \paren {a_3 a_4}$
 * $\quad \paren {a_1 \paren {a_2 a_3} } a_4$
 * $\quad \paren {\paren {a_1 a_2} a_3} a_4$

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 $n = 4$ we have: