  We consider maximum rooted tree extension counts in random graphs, i.e., we
consider M_n = \max_v X_v where X_v counts the number of copies of a given tree
in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the
random graph is not too sparse, specifically when the edge probability p=p(n)
satisfies p(1-p)n \gg \log n. The problem is more difficult in the sparser
regime 1 \ll pn \ll \log n, where we determine the asymptotics of M_n for
specific classes of trees. Interestingly, here our large deviation type
optimization arguments reveal that the behavior of M_n changes as we vary
p=p(n), due to different mechanisms that can make the maximum large.
