The number of ways to parenthesize an $n$ fold product is a Catalan number in the list $1,1,2,5,14,\cdots$ where these are in order of the number of terms in the product. The $n$th such number is also the numbr of ways to triangulate an $n+1$-gon.

I'm wondering whether there is a simple translation between a specific parenthesized $n$ fold product to a specific triangulation of an $n+1$-gon.

For example one parenthesized 5-fold product is $(12)(3(45))$ This then would (hopefully) be translatable to one of the $14$ triangulations of a $6$-gon.

I have tried various ways to label the vertices of the $n+1$ gon to see which parenthesized $n$ fold product a given triangulation goes with, but no luck.