Enumeration of multipartite series-reduced trees

TL;DR

This paper derives a generating function for rooted multipartite labeled series-reduced trees, enabling enumeration of various tree classes and related structures, and establishes connections with ultrametrics and chain-increasing binary trees.

Contribution

It provides a new generating function for degree sequences and colors of these trees, and links their enumeration to ultrametrics and other combinatorial structures.

Findings

Number of multipartite labeled series-reduced trees determined

Number of such trees equals that of colored chain-increasing binary trees

Refinement of Riordan and Shannon's results achieved

Abstract

We obtain a generating function for the degree sequences and colors of rooted multipartite labeled series-reduced trees. As an application of this result, we determine the number of symbolic ultrametrics (introduced by B\"ocker and Dress) and increasingly labeled processes. We also find that the number of multipartite labeled series-reduced trees and the colored chain-increasing binary trees are the same. We obtain the number of rooted multipartite unlabeled series-reduced trees. We also find a refinement of the result of Riordan and Shannon.