The main substance of the paper concerns the growth rate and the classificatio- n (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process, and leaves can be deleted at a rate . The main results lay the stress on the famous number e. In the case of a pure birth process, i.e. =0, the height of the tree at time t grows linearly at the rate e, in mean and almost surely as t. When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor =/¸: it is ergodic if e^-1, and transient if >e^-1. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jump. Bounds are obtained for the transient regime. Some basic stationary laws are computed, e.g. the number of nodes and the height. An extension to the so-called multiclass case is presented, with a more complex classification.