  Given a tree $T$ and a group $\Ga$ of automorphisms of $T$, we study the
markovian properties of the geodesic flow on the quotient by $\Ga$ of the space
of geodesics of $T$. For instance, when $T$ is the Bruhat-Tits tree of a
semi-simple connected algebraic group $\underline{G}$ of rank one over a non
archimedian local field $\wh K$, and $\Ga$ is a (possibly non uniform) lattice
in $\underline{G}(\wh K)$, we prove that the type preserving geodesic flow is
Bernoulli with finite entropy. Under some mild assumptions, we prove that if
the quotient geodesic flow is mixing for a probability
Patterson-Sullivan-Bowen-Margulis measure, then it is loosely Bernoulli.
