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

Let T n be the set of rooted labeled trees on { 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.