Let C(T) denote the poset of subtrees of a tree T with respect to the inclusion ordering. Jacobson, Kezdy and Seif gave a single example of a tree T for which C(T) is not Sperner, answering a question posed by Penrice. The authors then ask whether there exist an infinite family of trees T such that C(T) is not Sperner. This paper provides such a family.
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