[sagebird]

A more compact and beautiful relation exists between integers and finite rooted trees exist, imo.

David W. Matula found a correspondence between trees and integers using prime factorization, and reported it in 1968 in SIAM: "A Natural Rooted Tree Enumeration by Prime Factorization", SIAM Rev. 10, 1968, p.273 [1]

Others have commented on it before, search the web for Matula Numbers

I independently found this relation when working on a bar code system that was topologically robust to deformation. I wrote a document that explained this relation here[2].

I created an interactive javascript notebook that draws related topological diagrams for numbers. [3]

[1] http://williamsharkey.com/matulaSIAM.png

[2] https://williamsharkey.com/integer-tree-isomorphism.pdf

[3] https://williamsharkey.com/MatulaExplorer/MatulaExplorer.htm...

[sagebird]

Sorry - I believe I am off topic as this is not relevant given:

"This indirectly enforces the idea that sets cannot have duplicate elements, as set membership is defined purely by the presence or absence of elements. For example:"

So there is a constraint on what sort of trees are allowed in this -forrest- which would preclude most finite rooted trees.

[layer8]

From [2]:

> EG: 165 = P5 * P3 * P1

Shouldn’t the last component be P2 (= 3)?

[sagebird]

You are exactly correct - thank you for reading and letting me know, appreciate your curiosity!