  The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth
of a supercritical branching process, showing that an L log L condition is
decisive. In critical and subcritical cases, results of Kolmogorov and later
authors give the rate of decay of the probability that the process survives at
least n generations. We give conceptual proofs of these theorems based on
comparisons of Galton-Watson measure to another measure on the space of trees.
This approach also explains Yaglom's exponential limit law for conditioned
critical branching processes via a simple characterization of the exponential
distribution.
