Quick other one: To parse infix expressions, every time you see "x·y | (z | w)", find the operator of least binding power: In my example, I've given "|" less binding power than "·". Anyway, this visually breaks the expression into two halves: "x·y" and "(z | w)". Recursively parse those two subexpressions. Essentially, that's it.

The symbols "·" and "|" don't mean anything - I've chosen them to be visually intuitive: The "|" is supposed to look like a physical divider. Also, bracketed expressions "(...)" or "{...}" should be parsed first.

Wikipedia mentions that a variant of this got used in FORTRAN I. You could also speed up my naive O(n^2) approach by using Cartesian trees, which you can build using something suspiciously resembling precedence climbing.