Polyhedral CAT(0) metrics on locally finite complexes

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.