Sentences of monadic second order logic can express some quite involved properties of structures. Having a limit law, gives a uniform proof that all such properties have well defined asymptotic probabilities. One place this works is if the structures have ``components'' (which often just means connected components in the graph theory sense), and the numbers of structures and component structures have the ``right'' asymptotics. This is a somewhat different situation from the 0-1 laws for relational structures, where a typical structure has only one component. For example, a single unary function typically has many components, while (for an appropriate definition of ``component'') a rooted tree, on average, has a few.

LECTURE 1. Components. Fraisse-Ehrenfeucht games, as they apply to monadic second order (MSO) logic, and structures having components, will be reviewed. Combinatorial methods for counting structures with components by means generating series will then be sketched, followed by Compton's idea of using Tauberian lemmas to determine the asymptotic probability of a sentence. An example is a description of what a random unary function looks like, from an MSO perspective.

LECTURE 2. Trees. The asymptotics of finite trees will be sketched. The ideas will then be generalized to a multidimensional version which allows a limit law to be proved for all MSO sentences about a tree, and has other interesting applications.