Nondeterministic tree-walking automata are not closed under complementation

It is proved that the family of tree languages recognized by nondeterministic tree-walking automata is not closed under complementation, solving a problem raised by Bojańczyk and Colcombet (“Tree-walking automata do not recognize all regular languages”, SIAM J. Comp. 38 (2008) 658–701). In addition, it is shown that nondeterministic tree-walking automata are stronger than unambiguous tree-walking automata.