  Random spanning trees are among the most prominent determinantal point
processes. We give four examples of random spanning trees on ladder-like graphs
whose rungs form stationary renewal processes or regenerative processes of
order two, respectively. Up to a trivial thinning with additional coin flips,
for each of the first two examples the renewal processes exhaust the whole
class of stationary regenerative (of order one) determinantal point processes.
We also give an example of a regenerative process of order two that has no
representation in terms of a random spanning tree.
  Our examples illustrate a theorem of Lyons and Steif (2003) which
characterizes regenerative determinantal point processes in terms of their
Fourier transform. For the regenerative process, we also establish a Markov
chain description in the spirit of H\"aggstr\"om (1994).
  On the technical side, a systematic counting scheme for random spanning trees
is developed that allows to compute explicitly the probabilities. In some cases
an electrical network point of view simplifies matters.
