  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 determined 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.
