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 $n\geq 4$, 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.