  The theory of finite-size scaling explains how the singular behavior of
thermodynamic quantities in the critical point of a phase transition emerges
when the size of the system becomes infinite. Usually, this theory is presented
in a phenomenological way. Here, we exactly demonstrate the existence of a
finite-size scaling law for the Galton-Watson branching processes when the
number of offsprings of each individual follows either a geometric distribution
or a generalized geometric distribution. We also derive the corrections to
scaling and the limits of validity of the finite-size scaling law away the
critical point. A mapping between branching processes and random walks allows
us to establish that these results also hold for the latter case, for which the
order parameter turns out to be the probability of hitting a distant boundary.
