Chomsky hierarchy for tree structures
For tree automata, you have the Mostowski hierarchy, which is about the complexity of acceptance condition: each level is of the form $(i,j)$ with $i\in\{0,1\}$ and $i\leq j$. Being at level $(i,j)$ means that there is a parity automaton using parities from $i$ to $j$ recognizing the language. For more on parity condition, see here: https://en.wikipedia.org/wiki/%CE%A9-automaton. This hierarchy exists in 3 versions: deterministic, nondeterministic, or alternating, depending on the model of automaton you are looking at (although the deterministic hierarchy does not cover all regular languages).

There is an other hierarchy called Wadge hierarchy, with a more topological flavour, but it is getting further from complexity of acceptors: http://en.wikipedia.org/wiki/Wadge_hierarchy