  For a local field $F$ and an Artinian local coefficient ring $\Lambda$ with
the same positive residue characteristic $p$ we define, for any $e\in{\mathbb
N}$, a category ${\mathfrak C}^{(e)}(\Lambda)$ of ${\rm GL}_2(F)$-equivariant
coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is
an obvious functor from the category of ${\rm GL}_2(F)$-representations over
$\Lambda$ to ${\mathfrak C}^{(e)}(\Lambda)$. If $F={\mathbb Q}_p$ then
${\mathfrak C}^{(1)}(\Lambda)$ is equivalent to the category of smooth ${\rm
GL}_2({\mathbb Q}_p)$-representations over $\Lambda$ generated by their
invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show
that the subcategory of all objects in ${\mathfrak C}^{(e)}(\Lambda)$ with
trivial central character is equivalent to a category of representations of a
certain subgroup of ${\rm Aut}(X)$ consisting of "locally algebraic
automorphisms of level $e$". For $e=1$ there is a functor from this category to
that of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is a
bijection between irreducible objects. Finally, we present a parallel of
Colmez' functor $V\mapsto {\bf D}(V)$: to objects in ${\mathfrak
C}^{(e)}(\Lambda)$ (for any $F$) we assign certain \'{e}tale
$(\varphi,\Gamma)$-modules over an Iwasawa algebra ${\mathfrak
o}[[\widehat{N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawa
algebra ${\mathfrak o}[[{N}_{0}]]$. This assignment preserves finite
generation.
