In an article by Tom Davis on Catalan numbers, Davis mentions several problems that are shown to be equivalent. One of them is the problem of determining the number of strings of $n$ pairs of balanced parentheses. Another is the problem of determining the number of "multiplication orderings" of $n + 1$ numbers. In both cases, the answer to the problem is $C_n$, or the nth Catalan number.
For example, with $n = 3$, there are $C_3 = 5$ different strings of 3 pairs of balanced parentheses:
()()(), ()(()), (())(), (()()), ((()))
With four numbers, there are also $C_3$ multiplication orderings of them:
(((a·b)·c)·d), ((a·b)·(c·d)), ((a·(b·c))·d), (a·((b·c)·d)), (a·(b·(c·d)))
Davis shows how to convert a multiplication ordering of $n + 1$ terms to a string of $n$ pairs of balanced parentheses, but I would like to know how to go the other way; i.e. given a string of $n$ pairs of balanced parentheses, how can I calculate a corresponding multiplication ordering?