On the maximum mean subtree order of trees

TL;DR

This paper investigates the properties of trees with maximum mean subtree order, providing bounds and characterizations, and offers evidence supporting the conjecture that caterpillars maximize this invariant.

Contribution

It proves bounds on the maximum mean subtree order and characterizes trees like brooms and caterpillars as extremal structures, advancing understanding of this invariant.

Findings

Maximum mean subtree order is approximately n - 2 log2 n.

Trees with maximum mean subtree order have diameters close to n.

Brooms maximize local mean subtree order.

Abstract

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we do not completely resolve this conjecture, we find some evidence in its favor by proving different features of trees that attain the maximum. For example, we show that the diameter of a tree of order $n$ with maximum mean subtree order must be very close to $n$. Moreover, we show that the maximum mean subtree order is equal to $n - 2\log_2 n + O(1)$. For the local mean subtree order, which is the average order of all subtrees containing a fixed vertex, we can be even more precise: we show that its maximum is always attained by a broom and that it is equal to $n - \log_2 n + O(1)$.