  Aldous-Broder algorithm is a famous algorithm used to sample a uniform
spanning tree of any finite connected graph G, but it is more general: it
states that given a reversible M Markov chain on G started at r, the tree
rooted at r formed by the steps of successive first entrance in each node
(different from the root) has a probability proportional to
$\prod_{e=(e1,e2)\in{\sf Edges}(t,r)} M_{e1,e2}$ , where the edges are directed
toward r. As stated, it allows to sample many distributions on the set of
spanning trees. In this paper we extend Aldous-Broder theorem by dropping the
reversibility condition on M. We highlight that the general statement we prove
is not the same as the original one (but it coincides in the reversible case
with that of Aldous and Broder). We prove this extension in two ways: an
adaptation of the classical argument, which is purely probabilistic, and a new
proof based on combinatorial arguments. On the way we introduce a new
combinatorial object that we call the golf sequences.
