Tree languages defined in first-order logic with one quantifier alternation

International audienceWe study tree languages that can be defined in Δ2. These are tree languages definable by a first-order formula whose quantifier prefix is ∃ * ∀ * , and simultaneously by a first-order formula whose quantifier prefix is ∀ * ∃ *. For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in Δ2. This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11]