  In this paper we study several aspects of the growth of a supercritical
Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena
determined by the Schroder constant. We develop the local limit theory of Z_n,
that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to
study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an
LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n.
