State complexity of Kleene-star operations on regular tree languages

The concatenation of trees can be defined either as a sequential or a parallel operation, and the corresponding iterated operation gives an extension of Kleene-star to tree languages. Since the sequential tree concatenation is not associative, we get two essentially different iterated sequential concatenation operations that we call the bottom-up star and top-down star operation, respectively. We establish that the worst-case state complexity of bottom-up star is (n + 3/2) · 2^n-1. The bound differs by an order of magnitude from the corresponding result for string languages. The state complexity of top-down star is similar as in the string case. We consider also the state complexity of the star of the concatenation of a regular tree language with the set of all trees.