  In proper hyperbolic geodetic spaces we construct rooted $\mathbb R$-trees
with the following properties. On the one hand, every ray starting at the root
is quasi-geodetic; so these $\mathbb R$-trees represent the space itself well.
At the same time, the trees boundary reflects the boundary of the space in that
the number of disjoint rays to a boundary point is bounded in terms of the
(Assouad) dimension of the hyperbolic boundary.
