Higher-order recursion schemes are an expressive formalism used to define languages of finite and infinite ranked trees. They extend regular and context-free grammars, and are equivalent in expressive power to the simply typed λ Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity

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... tomata over infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes.