  A leaf path language is a Boolean combination of sets of the form
$\mathsf{{}^mE}^k L$, with $k \ge 1$ and $L$ a regular word language, which
consist of those forests where the node labels in at least $k$ leaf-to-root
paths make up a word that belongs to $L$. We look at the class $\mathsf{*D}$ of
the languages recognized by iterated wreath products of syntactic algebras of
leaf path languages. We prove the existence of an algorithm that, given a
regular forest language, returns in finite time a sequence of such algebras;
their wreath product is divided by the language's syntactic algebra if, and
only if this language belongs to $\mathsf{*D}$, which makes membership in this
class a decidable question. The result also applies to the subclasses
$\mathsf{PDL}$ and $\mathsf{CTL^*}$.
