In this lecture, we are interested in studying the asymptotic contour process of a GatlonWatson tree with Geometric(1/2) offspring distribution conditioned to be large. Instead of
looking at the contour process of a single tree, it will be easier to first consider an infinite
forest of trees. We will use Donsker’s theorem to find the limiting contour process of the
forest and then we will look for large trees that occur naturally in the infinite forest. In
order to simplify our lives, we will consider a signed forest. That is, we let F = ((Tk, Uk))k≥1
be an i.i.d sequence such that Tk and Uk are independent, P(Uk = 1) = P(Uk = −1) = 1/2
and Tk is a planted Gatlon-Watson tree with Geometric(1/2) offspring distribution, planted
meaning the root is conditioned to have degree equal to 1. We will call F a signed forest of
planted Gatlon-Watson trees with Geometric(1/2) offspring distribution.