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A Helly family of order $k$ is a set family for which every finite subfamily has nonempty total intersection whenever each of its subfamilies with at most $k$ members has nonempty intersection. Convex subtrees of a tree have Helly order two.

Finite families of convex subtrees of a tree have the Helly property: if every pair intersects, then their total intersection is nonempty.

In the terminology of a Helly family, convex subtrees form a family of Helly order two.