  We consider the satisfiability problem for the two-variable fragment of
first-order logic over finite unranked trees. We work with signatures
consisting of some unary predicates and the binary navigational predicates
child, right sibling, and their respective transitive closures. We prove that
the satisfiability problem for the logic containing all these predicates is
EXPSPACE-complete. Further, we consider the restriction of the class of
structures to singular trees, i.e., we assume that at every node precisely one
unary predicate holds. We observe that the full logic and even for unordered
trees remain EXPSPACE-complete over finite singular trees, but the complexity
decreases for some weaker logics. Namely, the logic with one binary predicate,
descendant is NEXPTIME-complete, and its guarded version is PSPACE-complete
over finite singular trees, even though both these logics are EXPSPACE-complete
over arbitrary finite trees.
