A Note on Weighted Rooted Trees

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 $ω$ be a nonnegative weight function defined on $V(T)$ with $\sum_{v\in V(T)}ω(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\sum_{v\in V(P)}ω(v)\ge \frac13$ for some $u\in V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in A }ω(a)\ge \frac13$ and $\sum_{b\in B }ω(b)\ge \frac13$. The bound $\frac13$ is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé.

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