On the finite degree of ambiguity of finite tree automata

The degree of ambiguity of a finite tree automaton A, da(A), is the maximal number of different accepting computations of A for any possible input tree. We show: it can be decided in polynomial time whether or not da(A)<∞. We give two criteria characterizing an infinite degree of ambiguity and derive the following fundamental properties of an finite tree automaton A with n states and rank L>1 having a finite degree of ambiguity: for every input tree t there is a input tree t₁ of depth less than 2²ⁿ·n! having the same number of accepting computations; the degree of ambiguity of A is bounded by 2^{22·log(L+1)·n}.