  An automaton is unambiguous if for every input it has at most one accepting
computation. An automaton is k-ambiguous (for k > 0) if for every input it has
at most k accepting computations. An automaton is boundedly ambiguous if it is
k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely
(respectively, countably) ambiguous if for every input it has at most finitely
(respectively, countably) many accepting computations.
  The degree of ambiguity of a regular language is defined in a natural way. A
language is k-ambiguous (respectively, boundedly, finitely, countably
ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly,
finitely, countably ambiguous) automaton. Over finite words every regular
language is accepted by a deterministic automaton. Over finite trees every
regular language is accepted by an unambiguous automaton. Over $\omega$-words
every regular language is accepted by an unambiguous B\"uchi automaton and by a
deterministic parity automaton. Over infinite trees Carayol et al. showed that
there are ambiguous languages.
  We show that over infinite trees there is a hierarchy of degrees of
ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1
ambiguous; and there are finitely (respectively countably, uncountably)
ambiguous languages that are not boundedly (respectively finitely, countably)
ambiguous.
