The Minimal Automorphism-Free Tree

A finite tree $T$ with $\mathrm{\mid }V(T)\mathrm{\mid }\ge 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{A}\mathcal{F}\mathcal{T}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1⪯T_2$ 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 $\mathcal{A}\mathcal{F}\mathcal{T}$ has a unique minimal element. This result gives an affirmative answer to the question asked by Rupinski.