  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.
