Passing memes between social platforms (and posting in October)...

Let be the set of rooted binary trees, where we include the empty tree with 0 nodes. There is an obvious bijection between nonempty trees in and pairs of trees in , by taking the left and right subtree of . So there is an obvious bijection between and {the empty tree, plus pairs of trees in }.

Let's write this as , so . In other words, is a sixth root of unity.

So . Unfortunately this is absurd because there is more than one 6-tuple of trees.

But, let's try again and write . And in fact:

• There is a "very explicit" bijection between the set of 7-tuples of trees, and the set of all trees. Here "very explicit" means that to map 7 trees to 1, you just need to inspect the 7 trees down to a fixed finite depth, independent of what the trees actually are, to construct some part of the 1 tree. Then you can paste the subtrees beneath that depth to subtrees of what you constructed.
• This is tight: there exists a very explicit bijection between the set of -tuples of trees and trees iff .

Now the question is, of course: how did treating , an infinite set of combinatorial objects, as a complex number produce a reasonable theorem?

It turns out there's some ring theory you can pull to see why this is the case. Some arXiv papers on the subject: https://arxiv.org/pdf/math/9405205v1.pdf https://arxiv.org/abs/math/0212377