Tree-walking automata do not recognize all regular languages

Tree-walking automata are a natural sequential model for recognizing tree languages. Every tree language recognized by a tree-walking automaton is regular. In this paper, we present a tree language which is regular but not recognized by any (nondeterministic) tree-walking automaton. This settles a conjecture of Engelfriet, Hoogeboom and Van Best. Moreover, the separating tree language is definable already in first-order logic over a signature containing the left-son, right-son and ancestor relations.