Maximizing the mean subtree order

TL;DR

This paper characterizes the structure of trees with maximum mean subtree order within specific families, introduces the Gluing Lemma, and resolves open questions about optimal trees and their leaf configurations.

Contribution

It establishes the Gluing Lemma, describes the leaf structure of optimal trees, and provides new insights into the asymptotic structure of optimal trees in various families.

Findings

Optimal trees have leaves adjacent to vertices of degree at least 3.

Number of leaves in optimal trees grows logarithmically with n.

Path graphs have minimum mean subtree order among all trees.

Abstract

This article focuses on properties and structures of trees with maximum mean subtree order in a given family; such trees are called optimal in the family. Our main goal is to describe the structure of optimal trees in $\mathcal{T}_n$ and $\mathcal{C}_n$, the families of all trees and caterpillars, respectively, of order $n$. We begin by establishing a powerful tool called the Gluing Lemma, which is used to prove several of our main results. In particular, we show that if $T$ is an optimal tree in $\mathcal{T}_n$ or $\mathcal{C}_n$ for $n\geq 4$, then every leaf of $T$ is adjacent to a vertex of degree at least $3$. We also use the Gluing Lemma to answer an open question of Jamison, and to provide a conceptually simple proof of Jamison's result that the path $P_n$ has minimum mean subtree order among all trees of order $n$. We prove that if $T$ is optimal in $\mathcal{T}_n$, then the number of leaves in $T$ is $\mathrm{O}(\log_2 n)$, and that if $T$ is optimal in $\mathcal{C}_n$, then the number of leaves in $T$ is $\mathrm{\Theta}(\log_2 n)$. Along the way, we describe the asymptotic structure of optimal trees in several narrower families of trees.