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

Let TnTn be the set of rooted labeled trees on {0,…,n}{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 Tn,kTn,k of TnTn, which is the set of rooted labeled trees whose maximal decreasing subtree has k+1k+1 vertices.