As someone who has never heard of most of these concepts before (plane trees, catalan numbers, ballot sequences, depth vectors), I found the question "Can you think of a way to efficiently generate a random plane tree?" confusing, and I only understood the problem being solved by first trying to understand the solution. After reading through, it seems like it's asking about generating a random plane tree drawn from a uniform distribution of all possible plane trees with a given number of nodes? Cool idea once I understood it though!