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July 7, 2025

How to Count n-Ary Trees

Posted by John Baez

How do you count rooted planar nn-ary trees with some number of leaves? For n=2n = 2 this puzzle leads to the Catalan numbers. These are so fascinating that the combinatorist Richard Stanley wrote a whole book about them. But what about n>2n \gt 2?

I’ll sketch one way to solve this puzzle using generating functions. This will give me an excuse to talk a bit about something called ‘Lagrange inversion’.

By the way, a mentally ill person may post lots of strange comments here under various names — perhaps even under my name. Please don’t engage with him, or even discuss this issue here, since it will only worsen his illness. Just ignore him and talk about the math.

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