Infinite Systems of Functional Equations and Gaussian Limiting Distributions

Systems of functional equations for generating functions appear in many combinatorial enumeration problems, for example in tree enumeration problems or in the enumeration of planar graphs (and related problems), see Drmota (2009). Usually, these enumeration techniques can be extended to take several parameters into account: the number of vertices, the number of edges, the number of vertices of a given degree etc. One of the simplest examples is that of rooted plane trees, that are defined as rooted trees, where each node has an arbitrary number of successors with a natural left-to-right-order. By splitting up at the root one obtains a recursive description of rooted plane trees (see Figure 1) which translates into corresponding relations for the counting generating function y(x) = ∑ n≥1 ynx : y(x) = x+ xy(x) + xy(x) + xy(x) + · · · = x 1− y(x) .