Inducibility and universality for trees

TL;DR

This paper investigates the limit densities of subtrees in trees, establishing bounds on inducibility for non-path/non-star trees, and constructs a universal sequence of trees with positive limit densities for all trees.

Contribution

It proves bounds on inducibility for certain trees and constructs a universal sequence of trees with positive densities for all subtrees, addressing open questions.

Findings

Non-path/non-star trees have inducibility at most 1 - ε1.

Existence of infinitely many trees with inducibility at least ε2.

Constructed a universal sequence of trees with positive limit densities.

Abstract

We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive $\varepsilon_1$ and $\varepsilon_2$ such that every tree that is neither a path nor a star has inducibility at most $1-\varepsilon_1$, where the inducibility of a tree $T$ is defined as the maximum limit density of $T$, and that there are infinitely many trees with inducibility at least $\varepsilon_2$. Finally, we construct a universal sequence of trees; that is, a sequence in which the limit density of any tree is positive.