Stochastically costed tree automata: Turakainen's theorem

We establish Turakainen's theorem in the framework of trees; precisely, we show that every forest of the form (t\C-A(t) > 0} (A an R-costed tree automaton) coincides with a stochastic forest, i.e. a forest of the form {t\C-P(t) > lambda} with P an actual double stochastic tree automaton and lambda a cut-point. As an application we obtain some remarkable closure properties concerning stochastic forests and languages.