  We prove the arborescence of any locally finite complex that is $CAT(0)$ with
a polyhedral metric for which all vertex stars are convex. In particular
locally finite $CAT(0)$ cube complexes or equilateral simplicial complexes are
arborescent. Moreover, a triangulated manifold admits a $CAT(0)$ polyhedral
metric if and only if it admits arborescent triangulations. We prove eventually
that every locally finite complex which is $CAT(0)$ with a polyhedral metric
has a barycentric subdivision which is arborescent.
