Surprisingly, there are combinatorial systems with just a single combinator[0]. Also, NAND along with a single axiom[0] can reproduce all of propositional calculus. By Curry-Howard-Labek, these are related facts and suggest that a universal tree likely exists in the tree calculus.

[0]:https://en.wikipedia.org/wiki/Iota_and_Jot

[1]:https://en.wikipedia.org/wiki/Nicod's_axiom