  Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a
descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$
of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are
unrelated. Let $\omega$ be a nonnegative weight function defined on $V(T)$ with
$\sum_{v\in V(T)}\omega(v)=1$. In this note, we prove that either there is an
$(r, u)$-path $P$ with $\sum_{v\in V(P)}\omega(v)\ge \frac13$ for some $u\in
V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in
A }\omega(a)\ge \frac13$ and $\sum_{b\in B }\omega(b)\ge \frac13$. The bound
$\frac13$ is tight. This answers a question posed in a very recent paper of
Bonamy, Bousquet and Thomass\'e.
