  Dendric shifts are defined by combinatorial restrictions of the extensions of
the words in their languages. This family generalizes well-known families of
shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval
exchange transformations. It is known that any minimal dendric shift has a
primitive $\mathcal{S}$-adic representation where the morphisms in
$\mathcal{S}$ are positive tame automorphisms of the free group generated by
the alphabet. In this paper we investigate those $\mathcal{S}$-adic
representations, heading towards an $\mathcal{S}$-adic characterization of this
family. We obtain such a characterization in the ternary case, involving a
directed graph with 2 vertices.
