Randomly evolving trees III

TL;DR

This paper investigates the properties of stationary randomly evolving trees, deriving equations for node distributions, and finds that linear, rod-like structures are more probable than highly branched trees.

Contribution

It provides exact solutions for node distribution equations in stationary trees using three specific offspring distributions, extending previous work.

Findings

Stationary trees tend to be rod-like rather than highly branched.

Exact probabilities for node counts are calculated for three distributions.

The evolution favors linear structures over complex branching.

Abstract

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.