A data structure for sets of trees.
You might want to check out g-tries. This is essentially the data structure you're looking for, but designed for use with general graphs instead of just trees. As such, I'm not sure that g-tries have good theoretical guarantees -- I think they use a graph canonization algorithm as a subroutine -- but in practice they seem to work well.

(Don't be scared that the linked paper is about "network motifs in biological networks": the g-trie is a perfectly good abstract data structure for graphs.)