Over the past few years, we ran a Seminar in Harmonic Analysis at the Mathematics Department of the University of Rome “La Sapienza”. In this seminar many of the talks given by staff members and visitors were concerned, directly or indirectly, with infinite trees or tree-like graphs, and their automorphism groups. Seminar notes were occasionally taken by one or both of us, and sometimes written up informally for distribution to newcomers to the seminar. After a while, we felt that it would be convenient to give a more coherent organization to these notes. Once this decision was taken it became apparent that, at the cost of some omission, the general aim of describing the group of automorphisms of a homogeneous tree and its irreducible unitary representations would provide a convenient focus which would include much of the material we had in mind. We felt that this approach would shed light on the connection between harmonic analysis on trees and harmonic analysis on hyperbolic spaces, by emphasizing the strict analogy between the group of automorphisms of the tree and real rank 1 semisimple Lie groups. This choice left out a lot of valuable material specifically concerning free groups and free products of finite groups. We felt however that the notes [F-T P2] and the memoir [F-T S2] could provide an introduction to these topics. We also decided not to treat the case of a semihomogeneous tree.