This paper is concerned with trees constructed with geodesic arcs in hyperbolic space. We shall be interested in determining how close to geodesics (in hyperbolic space) are the geodesics paths (within the tree), and also in bounding the Hausdorff dimension of the so-called rim of the tree. These estimates and bounds are very useful in studying the Hausdorff dimension of some limit sets appearing in the theory of Fuchsian groups and related to the asymptotic behaviour of geodesics of Riemann surfaces.