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By extending the breadth first search algorithm to any d-type critical orsubcritical irreducible branching forest, we show that such forests may beencoded through d independent, integer valued, d-dimensional random walks. Anapplication of this coding together with a multivariate extension of the BallotTheorem which is proved here, allow us to give an explicit form of the law ofthe total progeny, jointly with the number of subtrees of each type, in termsof the offspring distribution of the branching process. We then apply theseresults to some enumeration formulas of multitype forests with given degreesand to a new proof of the Lagrange-Good inversion Theorem.

Coding multitype branching forests: application to the law of the total progeny of branching forest and to enumerations