  It is known that all but finitely many leaves of a measured foliated
2-complex of thin type are quasi-isometric to an infinite tree with at most two
topological ends. We show that if the foliation is cooriented, and the
associated R-tree is self-similar, then a typical leaf has exactly one
topological end. We also construct the first example of a foliated 2-complex of
thin type whose typical leaf has exactly two topological ends. `Typical' means
that the property holds with probability one in a natural sense.
