  By extending the breadth first search algorithm to any d-type critical or
subcritical irreducible branching forest, we show that such forests may be
encoded through d independent, integer valued, d-dimensional random walks. An
application of this coding together with a multivariate extension of the Ballot
Theorem which is proved here, allow us to give an explicit form of the law of
the total progeny, jointly with the number of subtrees of each type, in terms
of the offspring distribution of the branching process. We then apply these
results to some enumeration formulas of multitype forests with given degrees
and to a new proof of the Lagrange-Good inversion Theorem.
