Definition:Parenthesization

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

Let $a_i$ denote elements of $S$.

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

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

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

Examples

 * $n=3$: $\quad a_1 \circ (a_2 \circ a_3)$, $(a_1 \circ a_2) \circ a_3$
 * $n=4$: $\quad a_1\circ(a_2\circ(a_3\circ a_4))$, $a_1\circ((a_2\circ a_3)\circ a_4)$, $(a_1\circ a_2)\circ(a_3\circ a_4)$, $(a_1\circ(a_2\circ a_3))\circ a_4$, $((a_1\circ a_2)\circ a_3)\circ a_4$