Diverse families of Random rooted trees A compilation of characteristics

This diploma thesis deals with four big groups of random trees, namely Polya trees, simple generated trees, increasing trees and scale-free trees. Di erent characteristics, similarities and di erences of these varieties are discussed, e.g. the limiting distribution of node-degrees. Most results are obtained using generating functions and methods of singulary analysis and stochastics. In the rst chapter the necessary background of stochastics and graph theory is given, which will become necessary throughout the work, knowledge of probability theory and analysis is favorable for the comprehension of the work. In the second chapter we discuss results tracing back to George Pólya and the year 1937. Based upon that we show that the limiting degreedistribution of Pólya trees is a normal distribution. The third chapter adresses simply generated trees, a group whose generating function ful lls a(z) = φ(a(z)), for a power series φ with nonnegative coe cients. This group is equivalent to the group of Galton-Watson trees, which correspond to a Galton-Watson branching process. We can obtain interesting results on the structure of those trees in context of Brownian excursions. In the fourth chapter we equip the trees with an additional parameter, namely the labelling of their nodes, and eye on those trees whose labellings along any path away from the root is increasing. For certain families of those increasing trees we can also nd limiting degree distributions. In the fth and last chapter we de ne graphs and trees, which are similar no networks occuring in the real world, but were discovered only recently, the Scale free graphs and trees. The marcant property of these trees is the development through growth, the limiting degree distribution is exponential and independent of the beginning structure of the graph.