  Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ for
a fixed $p>2$. Projective general linear groups
$G=\operatorname{PGL}_{2}(\mathbb{K})$ and
$H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on Bruhat-Tits
trees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of
$H$-subtrees $G.T_{H}$. Let $\Gamma$ be a Schottky subgroup such that
$\Gamma\backslash T_{G}$ is infinite volume and has an additional condition
named high-branchedness, and let $\Lambda$ be its limit set.
  We classify $\Gamma$-orbits in $G/H$. Let $C=g_{C}H\in G/H$. As a
generalization of Ratner's theorem, if $\Gamma\backslash g_{C}.T_{H}$ meets the
convex core of $\Gamma\backslash T_{G}$, then the $\Gamma$-orbit of $C$ is
either dense or closed in $ {\cal{C}}_{\Lambda}=\{g H:
\partial(g.T_{H})\cap\Lambda\neq\varnothing\}$.
