Counting branches in trees using games

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 ω-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 Niwiński.