We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart showed that in this setting one of the players always has a winning strategy, though it is not known in advance which player. We present simple necessary conditions for the first player to have a winning strategy, and establish an equivalence between winning sets that guarantee a win for the first player and maximal prefix codes. Using this equivalence, we derive a necessary algebraic condition for winning, and exhibit a family of games for which this algebraic condition is in fact equivalent to winning. We introduce the concept of coverings, and show that by covering the graph with an infinite labeled tree corresponding to the free group, we can derive a simple trait of maximal prefix codes.

翻译：我们研究在无限树上进行的双人交替移动博弈。主要关注树为完全（正则）且获胜集合为开放（相对于树上的乘积拓扑）的情形。Gale和Stewart证明在此设定下总有一方具有必胜策略，但事先无法确定是哪一方。我们提出了先手方具有必胜策略的简单必要条件，并建立了保证先手方获胜的获胜集合与极大前缀码之间的等价关系。利用该等价关系，我们推导出获胜的必要代数条件，并展示了一类博弈族，其代数条件实际上等价于获胜。我们引入了覆盖的概念，证明通过用对应于自由群的无限标记树覆盖图，可以推导出极大前缀码的简单特征。