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