  We show a new simple algorithm that solves the model-checking problem for
recursion schemes: check whether the tree generated by a given higher-order
recursion scheme is accepted by a given alternating parity automaton. The
algorithm amounts to a procedure that transforms a recursion scheme of order
$n$ to a recursion scheme of order $n-1$, preserving acceptance, and increasing
the size only exponentially. After repeating the procedure $n$ times, we obtain
a recursion scheme of order $0$, for which the problem boils down to solving a
finite parity game. Since the size grows exponentially at each step, the
overall complexity is $n$-EXPTIME, which is known to be optimal. More
precisely, the transformation is linear in the size of the recursion scheme,
assuming that the arity of employed nonterminals and the size of the automaton
are bounded by a constant; this results in an FPT algorithm for the
model-checking problem.
  Our transformation is a generalization of a previous transformation of the
author (2020), working for reachability automata in place of parity automata.
The step-by-step approach can be opposed to previous algorithms solving the
considered problem "in one step", being compulsorily more complicated.
