  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 theGW tree as it is just the number of
individuals at each generation. We then give alternativeproofs of classical
results on GW processes using the tree formalism. We focus in particular onthe
extinction probability (which was the first question of F. Galton) and on the
description ofthe processes conditioned on extinction or non extinction.In a
second chapter, we focus on local limits of conditioned GW trees. In the
critical andsub-critical cases, the population becomesa.s. extinct and the
associated genealogical tree is finite. However, it has a small but
positiveprobability of being large (this notion must be precised). The question
that arises is to describethe law of the tree conditioned of being large, and
to say what exceptional event has occurredso that the tree is not typical. A
first answer to this question is due to H. Kesten whoconditioned a GW tree to
reach height n and look at the limit in distribution when n tends toinfinity.
There are however other ways of conditioning a tree to be large: conditioning
on havingmany vertices, or many leaves... We present here very recent general
results concerning this kindof problems due to the authors of this course and
completed by results of X. He.
