Tree Languages Defined in First-Order Logic with One Quantifier Alternation

We study tree languages that can be defined in Δ2. These are tree languages definable by a first-order formula whose quantifier prefix is $\exists^*\forall^*$, and simultaneously by a first-order formula whose quantifier prefix is $\forall^*\exists^*$, both formulas over the signature with the descendant relation. We provide an effective characterization of tree languages definable in Δ2. This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ2forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11].