  Given a collection $\C$ of subsets of a finite set $X$, let $\bigcup \C =
\cup_{S \in \C}S$. Philip Hall's celebrated theorem \cite{hall} concerning
`systems of distinct representatives' tells us that for any collection $\C$ of
subsets of $X$ there exists an injective (i.e. one-to-one) function $f: \C \to
X$ with $f(S) \in S$ for all $S \in \C$ if and and only if $\C$ satisfies the
property that for all non-empty subsets $\C'$ of $\C$ we have $|\bigcup \C'|
\geq |\C'|$. Here we show that if the condition
  $|\bigcup \C'| \geq |\C'|$ is replaced by the stronger condition $|\bigcup
\C'| \geq |\C'|+2$, then we obtain a characterization of this condition for a
collection of 3-element subsets of $X$ in terms of the existence of an
injective function from $\C$ to the vertices of a tree whose vertex set
includes $X$ and that satisfies a certain median condition. We then describe an
extension of this result to collections of arbitrary-cardinality subsets of
$X$.
