  It is well known that for a regular tree language it is decidable whether or
not it can be recognized by a deterministic top-down tree automaton (DTA).
However, the computational complexity of this problem has not been studied. We
show that for a given deterministic bottom-up tree automaton it can be decided
in quadratic time whether or not its language can be recognized by a DTA. Since
there are finite tree languages that cannot be recognized by DTAs, we also
consider finite unions of \DTAs and show that also here, definability within
deterministic bottom-up tree automata is decidable in quadratic time.
