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

TL;DR

This paper extends coding methods for multitype branching forests using multidimensional random walks, enabling explicit calculations of total progeny distributions and applications to enumeration and inversion formulas.

Contribution

It introduces a novel coding of multitype branching forests via independent multidimensional random walks and derives explicit progeny laws using a multivariate Ballot Theorem extension.

Findings

Explicit law of total progeny in multitype forests

New coding method using multidimensional random walks

Applications to enumeration and inversion formulas

Abstract

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.