  It is well known that for a given bottom-up tree automaton it can be decided
whether or not there exists deterministic top-down tree automaton that
recognized the same tree language. Recently it was claimed that such a decision
can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their
procedure and corresponding property is wrong. Here we correct this mistake and
present a correct property which allows to determine in polynomial time whether
or not a given tree language can be recognized by a deterministic top-down tree
automaton. Furthermore, our new property is stated for arbitrary deterministic
bottom-up tree automata, and not for minimal such automata (as before).
