Forward triplets and topological entropy on trees

We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map \begin{document}$ f $\end{document} has positive entropy if and only if some iterate \begin{document}$ f^k $\end{document} has a periodic orbit with three aligned points consecutive in time, that is, a triplet \begin{document}$ (a,b,c) $\end{document} such that \begin{document}$ f^k(a) = b $\end{document} , \begin{document}$ f^k(b) = c $\end{document} and \begin{document}$ b $\end{document} belongs to the interior of the unique interval connecting \begin{document}$ a $\end{document} and \begin{document}$ c $\end{document} (a forward triplet of \begin{document}$ f^k $\end{document} ). We also prove a new criterion of entropy zero for simplicial \begin{document}$ n $\end{document} -periodic patterns \begin{document}$ P $\end{document} based on the non existence of forward triplets of \begin{document}$ f^k $\end{document} for any \begin{document}$ 1\le k inside \begin{document}$ P $\end{document} . Finally, we study the set \begin{document}$ \mathcal{X}_n $\end{document} of all \begin{document}$ n $\end{document} -periodic patterns \begin{document}$ P $\end{document} that have a forward triplet inside \begin{document}$ P $\end{document} . For any \begin{document}$ n $\end{document} , we define a pattern that attains the minimum entropy in \begin{document}$ \mathcal{X}_n $\end{document} and prove that this entropy is the unique real root in \begin{document}$ (1,\infty) $\end{document} of the polynomial \begin{document}$ x^n-2x-1 $\end{document} .