  Since the works of Howard and Newman (2001), it is known that in straight
radial rooted trees, with probability 1, infinite paths all have an asymptotic
direction and each asymptotic direction is reached by (at least) an infinite
path. Moreover, there exists a set of 'exceptionnal' directions reached by (at
least) two infinite paths which is random, dense and only countable in
dimension 2. Howard and Newman's method says nothing about (random) directions
reached by more than two infinite paths and, in particular, if such 'very
exceptionnal' directions exist in dimension 2. In this paper, we prove that the
answer is no for the hyperbolic Radial Spanning Tree (RST): in dimension 2,
this tree does not contain 3 infinite paths with the same (random) asymptotic
direction with probability one. Turned in another way, this means that there is
no infinite but thin subtree in the hyperbolic RST, i.e. whose infinite paths
would all have the same asymptotic direction. We actually prove a stronger
result in dimension $d+1$, $d\geq 1$, stating that any infinite subtree of the
hyperbolic RST a.s. generates a thick trace at infinity, i.e. the set of
asymptotic directions reached by its infinite paths has a positive measure.
