Paths vs. Stars in the Local Profile of Trees

Keywords: Trees, Subtrees, Local profile, Paths, Stars

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.