On the local profiles of trees

TL;DR

This paper investigates the local profile structures of trees, revealing that their limit sets of k-profiles form convex sets, and begins exploring the inequalities defining these convex sets, highlighting open problems.

Contribution

It demonstrates the convexity of the limit set of k-profiles of trees and initiates the study of their defining inequalities, contrasting with general graphs.

Findings

Limit set of k-profiles of trees is convex.

Initiates study of inequalities defining the convex set.

Many open problems remain in the area.

Abstract

We study the local profiles of trees. We show that, in contrast with the situation for general graphs, the limit set of k-profiles of trees is convex. We initiate a study of the defining inequalities of this convex set. Many challenging problems remain open.