Paths vs. stars in the local profile of trees

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 $p_1^{(k)}(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p_2^{(k)}(T)$ be the proportion of stars. Our main theorem states: if $p_1^{(k)}(T_n)\to 0$ for a sequence of trees $T_1,T_2,\dots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to 1$\mathrm{.}Botharealsoshowntobeequivalenttothestatementthatthenumberof$k$-vertexsubtreesgrowssuperlinearlyandthestatementthatthe$(k-1)$th degree moment grows superlinearly.