  In this article we study two related models of quantum geometry: generic
random trees and two-dimensional causal triangulations. The Hausdorff and
spectral dimensions that arise in these models are calculated and their
relationship with the structure of the underlying random geometry is explored.
Modifications due to interactions with matter fields are also briefly
discussed. The approach to the subject is that of classical statistical
mechanics and most of the tools come from probability and graph theory.
