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AFAIK there are many variations (I think infinite, even) of "tree calculi". You can build one easily from combinatory logic by using only one universal combinator, which will be implied at the leafs of the tree.

John Tromp in his repo has some research pertaining to short universal lambda expressions (each can be expressed as combinators) of this type: https://github.com/tromp/AIT/blob/master/Bases.lhs

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Super cool, thanks for the pointer! I'll note, though, that one of the main value adds of (this) TC is that it is also intensional.

See website for some elaboration and examples, I'd particularly recommend looking at https://treecalcul.us/live/?example=demo-fusion which demos a little stream fusion optimizer, entirely from scratch, including small test.

Oh, OK! I think it would be interesting to know to which universal combinator (or lambda expression) is the 't' related to.

Reduction rules (1) and (2) correspond to those of K and S, respectively. Reduction rule (3) corresponds to the elimination form for a Scott-encoding of a datatype like `X = Leaf | Stem X | Fork X X`. The cases of rule (3) are essentially parsing the argument into one of these three forms, and dispatching on the result.

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