  In this work we prove decidability of the model-checking problem for safe
recursion schemes against properties defined by alternating B-automata. We then
exploit this result to show how to compute downward closures of languages of
finite trees recognized by safe recursion schemes.
  Higher-order recursion schemes are an expressive formalism used to define
languages of finite and infinite ranked trees by means of fixed points of
lambda terms. They extend regular and context-free grammars, and are equivalent
in expressive power to the simply typed $\lambda Y$-calculus and collapsible
pushdown automata. Safety in a syntactic restriction which limits their
expressive power.
  The class of alternating B-automata is an extension of alternating parity
automata over infinite trees; it enhances them with counting features that can
be used to describe boundedness properties.
