  The class of Boolean combinations of tree languages recognized by
deterministic top-down tree automata (also known as deterministic
root-to-frontier automata) is studied. The problem of determining for a given
regular tree language whether it belongs to this class is open. We provide some
progress by two results: First, a characterization of this class by a natural
extension of deterministic top-down tree automata is presented, and as an
application we obtain a convenient method to show that certain regular tree
languages are outside this class. In the second result, it is shown that, for
fixed $k$, it is decidable whether a regular tree language is a Boolean
combination of $k$ tree languages recognized by deterministic top-down tree
automata.
