Connected subgraph of a tree

A nonempty connected subgraph of a tree. Between any two included vertices, it contains the unique tree path: its connectedness supplies a path, and the ambient tree has only one. Such subgraphs satisfy the Helly property. They need not contain every descendant of an included vertex in any chosen rooting, unlike the descendant-closed data-structure subtree convention.

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.