  Random-cluster measures on infinite regular trees are studied in conjunction
with a general type of `boundary condition', namely an equivalence relation on
the set of infinite paths of the tree. The uniqueness and non-uniqueness of
random-cluster measures are explored for certain classes of equivalence
relations. In proving uniqueness, the following problem concerning branching
processes is encountered and answered. Consider bond percolation on the
family-tree $T$ of a branching process. What is the probability that every
infinite path of $T$, beginning at its root, contains some vertex which is
itself the root of an infinite open sub-tree?
