Forest languages defined by counting maximal paths

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 $\mathsf{CTL^*}$.