Transport on randomly evolving trees.
The time process of transport on randomly evolving trees is investigated. By introducing the notions of living and dead nodes, a model of random tree evolution is constructed which describes the spreading in time of objects corresponding to nodes. It is assumed that at t=0 the tree consists of a single living node (root), from which the evolution may begin. At a certain time instant tau> or =0, the root produces v> or =0 living nodes connected by lines to the root which becomes dead at the moment of the offspring production. In the evolution process each of the new living nodes evolves further like a root independently of the others. By using the methods of the age-dependent branching processes we derive the joint distribution function of the numbers of living and dead nodes, and determine the correlation between these node numbers as a function of time. It is proved that the correlation function converges to square root of 3/2 independently of the distributions of v and tau when q1-->1 and t-->infinity. Also analyzed are the stochastic properties of the end nodes; and the correlation between the numbers of living and dead end nodes is shown to change its character suddenly at the very beginning of the evolution process. The survival probability of random trees is investigated and expressions are derived for this probability.