The present paper addresses the following question: for a geometric random tree in R-2, how many semi-infinite branches cross the circle C-r centered at the origin and with a large radius r? We develop a method ensuring that the expectation of the number X-r of these semi-infinite branches is o (r). The result follows from the fact that, far from the origin, the distribution of the tree is close to that of an appropriate directed forest which lacks bi-infinite paths. In order to illustrate its robustness, the method is applied to three different models: the Radial Poisson Tree (RPT), the Euclidean First-Passage Percolation (FPP) Tree and the Directed Last-Passage Percolation (LPP) Tree. Moreover, using a coalescence time estimate for the directed forest approximating the RPT, we show that for the RPT chi(r) is o (r(1) (eta)), for any 0 < eta < 1/4, almost surely and in expectation.