We study tree games developed recently by Matteo Mio as a game interpretation of the probabilistic $mu$-calculus. With expressive power comes complexity. Mio showed that tree games are able to encode Blackwell games and, consequently, are not determi
ned under deterministic strategies. We show that non-stochastic tree games with objectives recognisable by so-called game automata are determined under deterministic, finite memory strategies. Moreover, we give an elementary algorithmic procedure which, for an arbitrary regular language L and a finite non-stochastic tree game with a winning objective L decides if the game is determined under deterministic strategies.
