Maximizing the mean subtree order

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 $n\ge 4$, 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.

This version includes changes suggested by anonymous referees. Accepted to Journal of Graph Theory