Generic groups acting on regular trees

Let $T$ be a $k$-regular tree ($k\geq 3$) and $A=\mathrm{Aut}(T)$ its automorphism group. We analyze a generic finitely generated subgroup $\Gamma$ of $A$. We show that $\Gamma$ is free and establish a trichotomy on the closure $\overline{\Gamma }$ of $\Gamma$ in $A.$ It turns out that $\overline{\Gamma }$ is either discrete, compact or has index at most $2$ in $A$.