/ Enumeration of labeled trees

ʷ / Let ${\mathcal T}_{n}$ be the set of rooted labeled trees on

$\set{0,...,n}$. Given a rooted labeled tree $T$, A maximal decreasing

subtree of a rooted labeled tree is defined by the maximal subtree

from the root with all edges being decreasing. Recently Seo and Shin

have studied a new refinement ${\mathcal T}_{n,k}$ of ${\mathcal

T}_n$, which is the set of rooted labeled trees whose maximal

decreasing subtree has $k+1$ vertices. In this talk, we introduce the

refinement of the set of labeled ordered trees by the size of maximal

decreasing trees. We also present a refinement of the set of labeled

$p$-ary trees. This is a joint work with Heesung Shin.