This paper introduces the concept of *recognizable sets of finite graphs* and explores their properties using techniques from universal algebra and formal language theory. The main result states that every set of finite graphs definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. The paper also extends these results to unordered unbounded trees, showing that a set of such trees is recognizable if and only if it is definable in counting monadic second-order logic. Additionally, it proves that counting monadic second-order logic is strictly more powerful than ordinary monadic second-order logic in arbitrary logical structures. The paper concludes with applications to the analysis of recursive definitions and the decidability of the monadic second-order theory of context-free graph grammars.This paper introduces the concept of *recognizable sets of finite graphs* and explores their properties using techniques from universal algebra and formal language theory. The main result states that every set of finite graphs definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. The paper also extends these results to unordered unbounded trees, showing that a set of such trees is recognizable if and only if it is definable in counting monadic second-order logic. Additionally, it proves that counting monadic second-order logic is strictly more powerful than ordinary monadic second-order logic in arbitrary logical structures. The paper concludes with applications to the analysis of recursive definitions and the decidability of the monadic second-order theory of context-free graph grammars.