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Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodictree patterns with positive topological entropy and the set$\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible treepatterns. The aim of this paper is to determine the elements of minimum entropyin the families $\text{Pos}_n$, $\text{Irr}_n$ and$\text{Pos}_n\setminus\text{Irr}_n$. Let $\lambda_n$ be the unique real root ofthe polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct anirreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is$\log(\lambda_n)$. We prove that this entropy is minimum in $\text{Pos}_n$.Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ alsominimizes the entropy in the family $\text{Irr}_n$. We also prove that theminimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (whichis nonempty only for composite integers $n\ge6$) is $\log(\lambda_{n/p})/p$,where $p$ is the least prime factor of $n$.

Characterization of the tree cycles with minimum positive entropy for any period