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Easy Random Trees
Can you think of a way to efficiently generate a random plane tree? Richard P. Stanley in his book Catalan Numbers has a really nifty combinatorial proof of why Catalan numbers have the formula \[ C_n = {1 \over n+1}{2n \choose n} \] The standard proof uses generating functions applied to an inductive definition of the Catalan numbers, which frankly does little to illumiate their connection with combinatorial objects, despite the fact that the appearance of a binomial coefficient gives a hint that it’s counting something, and the division suggests we might be looking at some kind of equivalence class.
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