We use Dirichlet form methods to construct and analyse a general class of reversible Markov processes with totally disconnected state spaces. We study in detail the special case of bipartite Markov chains. The latter processes have a state space consisting of an ``interior'' with a countable number of isolated points and a, typically uncountable, ``boundary''. The equilibrium measure assigns all of its mass to the interior. When the chain is started at a state in the interior, it holds for an exponentially distributed amount of time and then jumps to the boundary. It then instantaneously re-enters the interior. There is a ``local time on the boundary''. That is, the set of times the process is on the boundary is uncountable and coincides with the points of increase of a continuous additive functional. Certain processes with values in the space of trees and the space of vertices of a fixed tree provide natural examples of bipartite chains. Moreover, time--changing a bipartite chain by its local time on the boundary leads to interesting processes, including particular L\'evy processes on local fields (for example, the $p$-adic numbers) that have been considered elsewhere in the literature.