  The present paper addresses the following question: for a geometric random
tree in $\R^{2}$, how many semi-infinite branches cross the circle $\CC_{r}$
centered at the origin and with a large radius $r$? We develop a method
ensuring that the expectation of the number $\chi_{r}$ 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.
