Definition:Parenthesization

Definition
Let $\circ$ be a product defined on a set $S$.

Let $a_i$ denote elements of $S$.

To distinguish all possible products of $a_1, a_2, \dotsc, a_n$ for some $n > 2$, parentheses are inserted into the product.

A set of parentheses applied on a product is called a parenthesization of that product.

Equivalent Parenthesizations
Two parenthesizations are equivalent the product defined by them yields the same result.

Examples

 * $n = 3$:
 * $\quad a_1 \circ \left({a_2 \circ a_3}\right)$
 * $\quad \left({a_1 \circ a_2}\right) \circ a_3$


 * $n = 4$:
 * $\quad a_1 \circ \left({a_2 \circ \left({a_3 \circ a_4}\right)}\right)$
 * $\quad a_1 \circ \left({\left({a_2 \circ a_3}\right) \circ a_4}\right)$
 * $\quad \left({a_1 \circ a_2}\right) \circ \left({a_3 \circ a_4}\right)$
 * $\quad \left({a_1 \circ \left({a_2 \circ a_3}\right)}\right) \circ a_4$
 * $\quad \left({\left({a_1 \circ a_2}\right) \circ a_3}\right) \circ a_4$