Paths vs. stars in the local profile of trees

TL;DR

This paper proves that in large trees, if the proportion of path subtrees diminishes, then the proportion of star subtrees approaches one, linking local subtree profiles to degree moments.

Contribution

It establishes a rigorous connection between the local profile of trees and degree moments, answering a recent open question by Bubeck and Linial.

Findings

If path subtrees vanish, star subtrees dominate in large trees.

Proportions are equivalent to superlinear growth of k-vertex subtrees.

Results link local subtree structure to degree distribution growth.

Abstract

The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p^{(k)}_2(T)$ be the proportion of stars. Our main theorem states: if $p^{(k)}_1(T_n) \to 0$ for a sequence of trees $T_1,T_2,\ldots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to 1$. Both are also shown to be equivalent to the statement that the number of $k$-vertex subtrees grows superlinearly and the statement that the $(k-1)$th degree moment grows superlinearly.