Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maximizing the mean subtree order   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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas