  We consider a model of random tree growth, where at each time unit a new
vertex is added and attached to an already existing vertex chosen at random.
The probability with which a vertex with degree $k$ is chosen is proportional
to $w(k)$, where the weight function $w$ is the parameter of the model.
  In the papers of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and,
independently, Mori, the asymptotic degree distribution is obtained for a model
that is equivalent to the special case of ours, when the weight function is
linear. The proof therein strongly relies on the linear choice of $w$.
  We give the asymptotical degree distribution for a wide range of weight
functions. Moreover, we provide the asymptotic distribution of the tree itself
as seen from a randomly selected vertex. The latter approach gives greater
insight to the limiting structure of the tree.
  Our proof relies on the fact that considering the evolution of the random
tree in continuous time, the process may be viewed as a general branching
process, this way classical results can be applied.
