  In this paper we introduce and study fuzzy deterministic top-down (DT) tree
automata over a lattice L. The L-fuzzy tree languages recognized by these
automata are said to be DT-recognizable, and they form a proper subfamily
$DRec_L$ of the family of $Rec_L$ of all regular L-fuzzy tree languages. We
prove a Pumping Lemma for $DRec_L$ from which several decidability results
follow. The closure properties of $DRec_L$ under various operations are
established. We also characterize DT-recognizability in terms of L-fuzzy path
languages, and prove that the path closure of any regular L-fuzzy tree language
is DT-recognizable, and that it is decidable whether a regular L-fuzzy tree
language is DT-recognizable. In most of the paper, L is just any nontrivial
bounded lattice, but sometimes it is assumed to be distributive or even a
bounded chain.
