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