The Brownian motion has played an important role in the development of probability theory and stochastic processes. We are going to see that it appears in the limiting process of several discrete processes. In particular, we will define discrete processes on Galton-Watson trees to see 2 different types of limits, which are the local limits and the scaling limits. The first result, Kesten's theorem, is a result for the local limits. We are going to look at the trees up to an arbitrary fixed height and therefore only consider what happens at a finite distance from the root. The second result concerns the limit of the rescaled height processes of an infinite Galton-Watson forest. We are going to consider sequences of trees where the branches are scaled by some factor so that all the vertices remain at finite distance from the root. Due to the scaling, the branches have infinitesimal length. These scaling limits lead to the so-called continuous random trees.

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