  We study numerically the critical behavior at the localization transition in
the Anderson model on infinite Bethe lattice and on random regular graphs. The
focus is on the case of coordination number $m+1 = 3$, with a box distribution
of disorder and in the middle of the band (energy $E=0$), which is the model
most frequently considered in the literature. As a first step, we carry out an
accurate determination of the critical disorder, with the result $W_c =18.17\pm
0.01$. After this, we determine the dependence of the correlation volume $N_\xi
= m^\xi$ (where $\xi$ is the associated correlation length) on disorder $W$ on
the delocalized side of the transition, $W < W_c$, by means of population
dynamics. The asymptotic critical behavior is found to be $\xi \propto
(W_c-W)^{-1/2}$, in agreement with analytical prediction. We find very
pronounced corrections to scaling, in similarity with models in high spatial
dimensionality and with many-body localization transitions.
