**Abstract** : The main object of this course given in Hammamet (December 2014) is the so-called Galton-Watson process.
We introduce in the first chapter of this course the framework of discrete random trees. We then use this framework to construct GW trees that describe the genealogy of a GW process. It is very easy to recover the GW process from the
GW tree as it is just the number of individuals at each generation. We then give alternative
proofs of classical results on GW processes using the tree formalism. We focus in particular on
the extinction probability (which was the first question of F. Galton) and on the description of
the processes conditioned on extinction or non extinction.
In a second chapter, we focus on local limits of conditioned GW trees. In the critical and
sub-critical cases, the population becomes
a.s. extinct and the associated genealogical tree is finite. However, it has a small but positive
probability of being large (this notion must be precised). The question that arises is to describe
the law of the tree conditioned of being large, and to say what exceptional event has occurred
so that the tree is not typical. A first answer to this question is due to H. Kesten who
conditioned a GW tree to reach height n and look at the limit in distribution when n tends to
infinity. There are however other ways of conditioning a tree to be large: conditioning on having
many vertices, or many leaves... We present here very recent general results concerning this kind
of problems due to the authors of this course and completed by results of X. He.