In the famous paper Seven trees in one, Andreas Blass showed that there is "a particularly elementary bijection between the set $T$ of finite binary trees and the set $T^7$ of seven-tuples of such trees".
For the Haskellers, if we define
data Tree = Leaf | Node Tree Tree
this leads to a bijection of types
iso :: Tree -> (Tree, Tree, Tree, Tree, Tree, Tree, Tree)
The justification, which this paper elaborates on, stems from the fact that for the set $T$ of trees, the recursive definitions yields an isomorphism $$T \cong T^2 + 1.$$ Treating this as an equation of complex numbers, we'd get
$$T = \frac 1 2 \pm \frac 1 2 i \sqrt 3$$
which is a primitive sixth root of unity, thus yielding $T^7 = T$.
This fascinating isomorphism just works for trees with no information attached to the nodes, as it just operates on the structure of the tree. I would love to see how we could incorporate actual, labelled nodes though (or better: colored nodes, see comments below). What do we get if we introduced labels on the nodes (with one of $n$ possibilities)? Say
data Label = A1 | ... | An
data Tree = Leaf | Node Label Tree Tree
Now we have an isomorphism $$T \cong 1 + nT^2$$ with complex solution $$T = \frac 1{2n}(1+i\sqrt{4n-1}).$$
E.g. for $n=2$ we had
$$T = \frac 1 4 (1 \pm i\sqrt 7).$$
None of these solutions can be roots of unity for $k>1$ because of their absolute values, so some nontrivial isomorphism $T^k \cong T^\ell$ won't arise. Are there other results one can achieve?
I know that this question basically amounts to asking if there is nice integral polynomial equation satisfied by the complex value for $T$ above, which means nice polynomial multiples of $nT^2-T+1$. Has there been any work done on these?
Thank you for your comments
I know this question, though it's way broader (and hasn't received an answer anyway).
Have a look at this paper. It gives some results for lists, naturals, and constants as well.