us VS ego

It might be easier to think about it as a stack, rather than a tree. Each element of the stack represents a subtree -- a perfect binary tree. If you ever have two subtrees of height k, you merge them together into one subtree of height k+1. Your stack might already have another subtree of height k+1; if so, you repeat the process, until there's at most one subtree of each height.

This process is isomorphic to binary addition. Worked example: let's start with a single leaf, i.e. a subtree of height 0. Then we "add" another leaf; since we now have a pair of two equally-sized leaves, we merge them into one subtree of height 1. Then we add a third leaf; now this one doesn't have a sibling to merge with, so we just keep it. Now our "stack" contains two subtrees: one of height 1, and one of height 0.

Now the isomorphism: we start with the binary integer 1, i.e. a single bit at index 0. We add another 1 to it, and the 1s "merge" into a single 1 bit at index 1. Then we add another 1, resulting in two 1 bits at different indices: 11. If we add one more bit, we'll get 100; likewise, if we add another leaf to our BNT, we'll get a single subtree of height 2. Thus, the binary representation of the number of leaves "encodes" the structure of the BNT.

This isomorphism allows you to do some neat tricks, like calculating the size of a Merkle proof in 3 asm instructions. There's some code here if that helps: https://github.com/lukechampine/us/blob/master/merkle/stack....

You could also check out section 5.1 of the BLAKE3 paper: https://github.com/BLAKE3-team/BLAKE3-specs/blob/master/blak...

This isn't what I'm asking for - I don't care if it's baked into us, exists as a backend for minio, uses PseudoKV https://github.com/lukechampine/us/issues/67, or whatever the case may be. I see no value in sending any third party my private data in an unencrypted form (uploading to your server, even if over HTTPS, you got my data).

Shameless plug, I also maintain an OCaml implementation of egraphs (named ego) at https://github.com/verse-lab/ego

While the most popular implementation at the moment seems to be egg in Rust, I find that OCaml serves as a much more ergonomic environment for quickly prototyping out uses of egraphs in practice. As a bonus, ego also shares the same logical interface as egg itself, so once you've finalised your designs, you shouldn't have much trouble porting them to egg if you need the performance gains.