  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.
