The subject of this thesis is the application of computer algebra systems, MAPLE in particular, to asymptotic problems coming from discrete mathematics. Two specific problems are examined.
The first is concerned with patterns in labeled trees. We count the average number of times a particular pattern matches a tree of size n. Assuming that every tree of size n is equally likely, it is shown that the limiting distribution of the number of occurrences of the pattern is asymptotically normal, with mean value [mu]*n and variance [sigma]2 [[sigma] hoch 2]*n with computable constants [mu]>0 and [sigma]>=0. We provide an algorithm to compute [mu] explicitly and a MAPLE program that does most of the work. This part of the thesis is based on the paper ``The Distribution of Patterns in Random Trees'' coauthored with Frédéric Chyzak and Michael Drmota.
In the second part of the thesis, we generalize a class of functions which were introduced by Hayman and which we thus call Hayman-admissible. Hayman proved that the suitably normalized coefficients of these functions asymptotically follow a Gaussian distribution. He also assembled a useful list of closure properties, i.e., operations on Hayman-admissible functions that generate other Hayman-admissible functions. We generalize Hayman's result to functions in two dimensions, conserving many closure properties. We also present a MAPLE program that tests if a given function belongs to this class. This part of the thesis is based on the paper ``Extended Admissible Functions and Gaussian Limiting Distributions'' coauthored with Michael Drmota and Bernhard Gittenberger.