  The properties of randomly evolving special trees having defined and analyzed
already in two earlier papers (arXiv:cond-mat/0205650 and
arXiv:cond-mat/0211092) have been investigated in the case when the continuous
time parameter converges to infinity. Equations for generating functions of the
number of nodes and end-nodes in a stationary (i.e. infinitely old) tree have
been derived. In order to solve exactly these equations we have chosen three
simple distributions for the number of new nodes produced by one dying node. By
using appropriate method we have calculated step-by-step the probabilities of
finding n=1,2,... nodes as well as end-nodes in a stationary random tree. The
conclusion to be correct that in the evolution process the formation of a
rod-like stationary tree is much more probable than the formation of a tree
with many branches.
