Equicontinuous mappings on finite trees

We show that, if $X$ is a finite tree (a dendrite with finitely many branching points, each of which has finite order; equivalently, a compact connected polyhedron without simple closed curves) and $f\colon X\longrightarrow X$ is a continuous function, then the following conditions are equivalent: (a) $f$ is equicontinuous (that is, the family of iterates of $f$ is an equicontinuous family of functions); (b) there is an $n\in\mathbb N$ such that the restriction of $f^n$ to $\bigcap_{m=1}^\infty f^m[X]$ is the identity function; (c) there exists an $n\in\mathbb N$ such that $\mathrm{Fix}(f^n)=\bigcap_{m=1}^\infty f^m[X]$ (where $\mathrm{Fix}(f^n)$ is the set of fixed points of $f^n$); (d) $\mathrm{Per}(f)=\bigcap_{m=1}^\infty f^m[X]$ (where $\mathrm{Per}(f)$ is the set of periodic points of $f$); (e) there is no arc $A\subseteq X$ satisfying $A\subsetneq f^n[A]$ for some $n\in\mathbb N$; (f) for every $n\in\mathbb N$, the set $\mathrm{Fix}(f^n)$ is connected; (g) the set $\mathrm{Per}(f)$ is connected; (h) for every nonprincipal ultrafilter $u$, the function $f^u\colon X\longrightarrow X$ is continuous; (i) for some nonprincipal ultrafilter $u$, the function $f^u\colon X\longrightarrow X$ is continuous. This generalizes a result of Vidal-Escobar and Garc\'ia-Ferreira (who proved the equivalence of (a), (e) and (h) in the case that $X$ is a $k$-od, $k\geq 3$), and complements earlier results of Bruckner and Ceder (dealing with the case where $X$ is an arc), Mai (in the case where $X$ is a finite graph) and Camargo, Rincon and Uzcategui (with $X$ being an arbitrary dendrite).