  This work is a study of the expressive power of unambiguity in the case of
automata over infinite trees. An automaton is called unambiguous if it has at
most one accepting run on every input, the language of such an automaton is
called an unambiguous language. It is known that not every regular language of
infinite trees is unambiguous. Except that, very little is known about which
regular tree languages are unambiguous.
  This paper answers the question whether unambiguous languages are of bounded
complexity among all regular tree languages. The notion of complexity is the
canonical one, called the (parity or Rabin-Mostowski) index hierarchy. The
answer is negative, as exhibited by a family of examples of unambiguous
languages that cannot be recognised by any alternating parity tree automata of
bounded range of priorities.
  Hardness of the given examples is based on the theory of signatures in parity
games, previously studied by Walukiewicz. This theory is further developed here
to construct canonical signatures. The technical core of the article is a
parity game that compares signatures of a given pair of parity games (without
an increase in the index).
