It is well known that for a given bottom-up tree automaton it can be decided whether or not an equivalent deterministic top-down tree automaton exists. 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 address 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 only for minimal such automata (as before).

众所周知，对于给定的自下而上树自动机，可以确定是否存在等效的确定性自上而下树自动机。最近有人声称这样的决策可以在多项式时间内完成（Leupold 和 Maneth，FCT'2021）；但他们的程序和相应的属性是错误的。在这里，我们解决了这个错误，并提出了一个正确的属性，它允许在多项式时间内确定给定的树语言是否可以被确定性自上而下的树自动机识别。此外，我们的新属性是针对任意确定性自下而上的树自动机，而不仅仅是针对最小的此类自动机（如之前）。