We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting. In this article, we relax the notion of accepting run by allowing a certain quantity of rejecting branches. More precisely we study the following criteria for a run to be accepting: (i) it contains at most finitely (\resp countably) many rejecting branches; (ii) it contains infinitely (\resp uncountably) many accepting branches; (iii) the set of accepting branches is topologically “big”. In all situations we provide a simple acceptance game that later permits to prove that the languages accepted by automata with cardinality constraints are always omega-regular. In the case (ii) where one counts accepting branches it leads to new proofs (without appealing to logic) of a result of Beauquier and Niwinski.