Chomsky hierarchy for tree structures I know of the Chomsky hierarchy, which concerns the expressive power of grammars to recognize languages $L \subseteq \Sigma^*$ made of words on an alphabet $\Sigma$.

Is there a similar hierarchy for grammars/parsers which are specifically taylored for nested words or trees?  In particular, this hierarchy should include tree automata, perhaps streaming tree automata, the algorithms used to apply CSS rules...

As I am planning to implement parsers for one/some of the steps in the hierarchy, I am very interested also in the algorithmic complexity of parsers for those grammars.  I might use this as a basis for a CSS-like functionality in LaTeX.
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