On the enumeration of rooted trees with fixed size of maximal decreasing trees

Let $T_{n}$ be the set of rooted labeled trees on $set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $T_{n,k}$ of $T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.