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The aim of this paper is to provide an affirmative answer to a recentquestion 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 proportionof stars. Our main theorem states: if $p^{(k)}_1(T_n) \to 0$ for a sequence oftrees $T_1,T_2,\ldots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to1$. Both are also shown to be equivalent to the statement that the number of$k$-vertex subtrees grows superlinearly and the statement that the $(k-1)$thdegree moment grows superlinearly.

Paths vs. stars in the local profile of trees