We have been running rec with some more complex 3- and 4-d trees taken from the figures in Wrapping of Trees. We have two automaton and accompanying trees recognizing the strings "Who does Carol think Alice does like?" and "It does seem that Alice does like Bob". We also tried to do a more generic automaton but we don't understand the theory behind the syntax so we weren't able to construct a grammar for it.
Automaton 1
Tree 1
Automaton 2
Tree 2