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Let $\\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1 \\preceq T_2$ if $T_1$ can be obtained from $T_2$ by successively deleting one leaf at a time in such a way that each intermediate tree is also automorphism-free. In this paper, we prove that $\\mathcal{AFT}$ has a unique minimal element. 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