  Directed covers of finite graphs are also known as periodic trees or trees
with finitely many cone types. We expand the existing theory of directed covers
of finite graphs to those of infinite graphs. While the lower growth rate still
equals the branching number, upper and lower growth rates do not longer
coincide in general. Furthermore, the behaviour of random walks on directed
covers of infinite graphs is more subtle. We provide a classification in terms
of recurrence and transience and point out that the critical random walk may be
recurrent or transient. Our proof is based on the observation that recurrence
of the random walk is equivalent to the almost sure extinction of an
appropriate branching process. Two examples in random environment are provided:
homesick random walk on infinite percolation clusters and random walk in random
environment on directed covers. Furthermore, we calculate, under reasonable
assumptions, the rate of escape with respect to suitable length functions and
prove the existence of the asymptotic entropy providing an explicit formula
which is also a new result for directed covers of finite graphs. In particular,
the asymptotic entropy of random walks on directed covers of finite graphs is
positive if and only if the random walk is transient.
