Asymptotic Analysis of Shape Parameters...

Trees and Lattice Paths are two inherently connected, elementary combinatorial structures. Within this thesis, tools from the Analytic Combinatorics framework are adapted and used in order to conduct an asymptotic analysis of certain shape parameters of interest.

Predominantly, the parameters under investigation are associated to deterministic reduction procedures. To be more precise, a suitable deterministic reduction naturally induces a notion of age on the objects (in the sense that "older" objects require more reductions until they are "irreducible"). Prominent examples of parameters that can be modeled by this approach include the height of plane trees, the register function (Horton--Strahler index) of binary trees, as well as the pruning index.

The extensive calculations involved in the asymptotic analysis of these shape parameters are carried out with the help of the free open-source computer mathematics software system SageMath and its included module for computations with asymptotic expansions, developed by Clemens Heuberger, Daniel Krenn, and the author. Besides straightforward verification of the stated results, this also allows to analyze the parameters with an unusually high degree of precision.