Someone recently gave me a random programming problem: to find the number of unique values of an infix expression (like say 1+2-3*4) parenthesized in all possible ways (in this example it's three: {-9 -3 0} because of the five ways two produce duplicate results). The operators were restricted to {+ - *}, the operands to positive integers. Somewhat arbitrary, but a fun puzzle... Anyway, I was wondering: is there a method for constructing such expressions such that over parenthesization they give the maximum possible Catalan(N) unique values? What is the maximum achievable if the operators are restricted to {+ *} or {- *}? Perhaps there are also interesting puzzles with restrictions on the operands, such as when they are all the same value, or powers of two, or the like...