FWF - Diameters - Random Recursive Structure of Small Diameters

Random Recursive Structures of Small Diameters Abstract The major aims of the AustrianTaiwanese joint project are: (1) to get a deeper understanding of the stochastic nature of random trees and more general structures which have a small diameter, (2) development and advancement of general analytic tools, (3) stimulating interdisciplinary research on an international level. Tree structures play an important role in many areas like computer science, quantum mechanics, biology and many more, and also in many subfields inside mathematics. They serve for instance as data structure. If one faces random data and stores them in a tree, then the resulting tree will be a random tree. In order to understand how this tree typically looks like (for instance for designing efficient algorithms for information retrieval which exploit the knowledge on the shape of the tree), it is necessary to consider large random trees and analyze them systematically. Trees may serve as a model of some real world phenomenon which we want to understand. In order to analyze a model, one starts with simple models, like e.g. branching processes. This model is very well understood, but its drawback is that there are many situations where it is not realistic. There are other models like binary search trees which are wellsuited for certain frameworks in computer science. Many of those new models have the property that the socalled diameter is small in comparison to the old models. In contrary to the branching process models, there exists no general framework for the new models, but only many partial results for many different models of trees with small diameter, Partial results indicate that such a framework can be described by means of Riccatilike differential equations. Our aim is to explore this phenomenon and to enhance existing methods for retrieving the desired information from these equations. A further goal of the project is to deepen the understanding of the typical shape of various classes of random trees. This is done by first studying the profile of random trees and then several shape parameters simultaneously. In applications one is often interested in the number of way to decompose a complex structure in distinct parts. This is a highly nontrivial problem, but we hope that the tools we develop during the project will enable us to obtain deep results on this question. In this context, there are examples of tree models from biology, but also other interesting structures from other fields of mathematics. Finally, a particular problem from information theory shall be investigated: If we do not focus on the tree size but on some other parameter, then this gives rise to a bias towards small diameter and probably trees with very different behaviour.