Randomly evolving trees III

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 different distributions for the number of new nodes ν produced by one dying node. By using appropriate method we have calculated step-bystep the probabilities of finding n = 1, 2, . . . nodes as well as end-nodes in a stationary random tree. Analyzing the results of numerical calculations we have observed that the qualitative properties of stationary random trees depend hardly on the character of distribution of ν. 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. We have established that the probability of finding n nodes in a stationary tree depends sensitively on the average value of ν and has a maximum the location of which is increasing with n but remains always smaller than unity. This is also true for the end-nodes. PACS: 02.50.-r, 02.50.Ey, 05.40.-a