  We present a new approach to the following meta-problem: given a quantitative
property of trees, design a type system such that the desired property for the
tree generated by an infinitary ground lambda-term corresponds to some property
of a derivation of a type for this lambda-term, in this type system.
  Our approach is presented in the particular case of the language finiteness
problem for nondeterministic higher-order recursion schemes (HORSes): given a
nondeterministic HORS, decide whether the set of all finite trees generated by
this HORS is finite. We give a type system such that the HORS can generate a
tree of an arbitrarily large finite size if and only if in the type system we
can obtain derivations that are arbitrarily large, in an appropriate sense; the
latter condition can be easily decided.
