Take x itself to be a bracketing, then recursively define a bracketing as a sequence B = (B_1, ..., B_k) where k>=2 and each B_i is a bracketing. A bracketing can be represented as a parenthesized string of xs, with parentheses removed from any single letter x for clarity of notation. Bracketings built up of binary operations only are called binary bracketings. For example, four letters have 11 possible bracketings: x x x x | (x x) x x | x(x x) x | x x(x x) (x x x) x | x(x x x) | ((x x) x) x | (x(x x)) x (x x)(x x) | x((x x) x) | x(x(x x)), | the last five of which are binary.

binary bracketing | Plutarch numbers | super Catalan number