On the difference of mean subtree orders under edge contraction

Abstract

Given a tree T of order n, one can contract any edge and obtain a new tree $T^{⁎}$ of order $n-1$. In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least $\frac{1}{3}$ in contracting an edge of a tree. In 2023, Luo, Xu, Wagner and Wang proved the case when the edge to be contracted is a pendant edge. In this article, we prove that the conjecture is true in general.