  We investigate the statistics of trees grown from some initial tree by
attaching links to preexisting vertices, with attachment probabilities
depending only on the valence of these vertices. We consider the asymptotic
mass distribution that measures the repartition of the mass of large trees
between their different subtrees. This distribution is shown to be a broad
distribution and we derive explicit expressions for scaling exponents that
characterize its behavior when one subtree is much smaller than the others. We
show in particular the existence of various regimes with different values of
these mass distribution exponents. Our results are corroborated by a number of
exact solutions for particular solvable cases, as well as by numerical
simulations.
