= Helly property for connected subgraphs of a tree
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A finite family of nonempty <connected subgraphs of a tree> with pairwise nonempty intersections has a common <vertex>. For a finite ambient <tree>, prove this by deleting a <leaf of a tree> $v$. If one member is $\{v\}$, pairwise intersection proves the result immediately. Otherwise remove $v$ from every member containing it. Each such member remains nonempty and connected and contains the unique neighbour of $v$. Hence every pair still intersects, even if its original intersection consisted only of $v$, and <mathematical induction> applies. This is the <Helly property> specialized to <trees>.
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