  The object of this paper is the study of a class of dessins d'enfants, the
so-called diameter four trees. These objects, first introduced by G. Shabat,
can be considered as the simplest non trivial example of etale covers of the
projective line minus three points. Their arithmetic properties are still
mysterious, and their study can inspire the understanding of more general
situations. Here, the main interest is devoted to the action of the absolute
Galois group on these (isomorphism classes of) coverings. In particular, in
many cases, we are able to distinguish Galois orbits and to describe the action
of the decomposition groups. One other central result concerns the study of
wild ramification, for which we show how to reduce to the tame case, and then
deduce some detailed arithmetical informations.
