In computer science, one way of providing a semantics for programs is by using infinite trees to model do-loops in finite flow-graphs.  Dana Scott and Marshall Hall are the earliest I recall.  Arbib and Manes later on.

A lot of knot theory can be tackled by a similar tack.

**Addenda**

For an introduction to asymptotic enumeration and random graphs (mentioned several times below), see:

* Edgar M. Palmer (1985), <i>Graphical Evolution : An Introduction to the Theory of Random Graphs</i>, John Wiley and Sons, New York, NY.  http://portal.acm.org/citation.cfm?id=4050

For one of the inaugural applications of graph theory to social networks, see:

* Frank Harary, Robert Z. Norman, and Dorwin Cartwright (1965), <i>Structural Models : An Introduction to the Theory of Directed Graphs</i>, Wiley, New York, NY.  http://www.ams.org/featurecolumn/archive/networks7.html

For recursive and self-similar graphs in knot theory, an ever-good springboard is:

* Louis H. Kauffman's home page : http://www.math.uic.edu/~kauffman/