  A non-deterministic recursion scheme recognizes a language of finite trees.
This very expressive model can simulate, among others, higher-order pushdown
automata with collapse. We show decidability of the diagonal problem for
schemes. This result has several interesting consequences. In particular, it
gives an algorithm that computes the downward closure of languages of words
recognized by schemes. In turn, this has immediate application to separability
problems and reachability analysis of concurrent systems.
