  Generating function equation has been derived for the probability
distribution of the number of nodes with $k \ge 0$ outgoing lines in randomly
evolving special trees. The stochastic properties of end-nodes (k=0) have been
analyzed, and it was shown that the relative variance of the number of
end-nodes vs. time has a maximum when the evolution is either subcritical or
supercritical. On the contrary, the time dependence of the relative dispersion
of the number of dead end-nodes shows a minimum at the beginning of the
evolution independently of its type. For the sake of better understanding of
the evolution dynamics the survival probability of random trees has been
investigated, and asymptotic expressions have been derived for this probability
in the cases of subcritical, critical and supercritical evolutions. In critical
evolution it was shown that the probability to find the tree lifetime larger
than x, is decreasing to zero as 1/x, if x tends to infinity. Approaching the
critical state it has been found the fluctuations of the tree lifetime to
become extremely large, and so near the critical state the average lifetime
could be hardly used for the characterization of the process.
