Nilpotent subsemigroups of a semigroup of order-decreasing transformations of a rooted tree

This paper deals with a semigroup of order-decreasing transformations of a rooted tree. Such are the transformations $\alpha$ of some rooted tree $G$ satisfying following condition: for any $x$ from $G$ $\alpha(x)$ belongs to a simple path from $x$ to the root vertex of $G$. We describe all subsemigroups of the mentioned semigroup, which are maximal among nilpotent subsemigroups of nilpotency class $k$ in our semigroup. In the event when rooted tree is a ray we prove that all these maximal subsemigroups are pairwise nonisomorphic.