Further results on the inducibility of $d$-ary trees

TL;DR

This paper investigates the inducibility of $d$-ary trees, providing explicit calculations for certain cases, establishing bounds, and analyzing the convergence rate of maximum densities in strictly $d$-ary trees.

Contribution

It explicitly determines inducibility in previously unknown cases, establishes bounds for balanced trees, and analyzes convergence rates supporting an open conjecture.

Findings

Explicit inducibility values for certain trees

General upper and lower bounds for inducibility

Convergence rate analysis of maximum density in $d$-ary trees

Abstract

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree $D$ inside a larger tree $T$ is the proportion of such leaf-induced subtrees in $T$ that are isomorphic to $D$ among all those with the same number of leaves as $D$. The inducibility of $D$ measures how large this density can be as the size of $T$ tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where $D$ is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of $D$ in strictly $d$-ary trees $T$ (trees where every internal vertex has precisely $d$ children) of a given size $n$ to the inducibility as $n \to \infty$, which supports an open conjecture.