  We study two models of an age-biased graph process: the $\delta$-version of
the preferential attachment graph model (PAM) and the uniform attachment graph
model (UAM), with $m$ attachments for each of incoming vertices. We show that
almost surely the scaled size of a breadth-first (descendant) tree rooted at a
fixed vertex converges, for $m=1$, to a limit whose distribution is a mixture
of two beta-distributions and a single beta-distribution respectively, and that
for $m>1$ the limit is $1$. We also analyze the likely performance of two
greedy (online) algorithms, for a large matching set and a large independent
set, and determine--for each model and each greedy algorithm--both a limiting
fraction of vertices involved and an almost sure convergence rate.
