Breadth first search coding of multitype forests with application to Lamperti representation

TL;DR

This paper establishes a bijection between multidimensional sequences and multitype plane forests using breadth-first search, linking discrete and continuous multitype branching processes via Lamperti transformations.

Contribution

It introduces a new coding method for multitype forests and extends Lamperti transformations to multitype branching processes in both discrete and continuous time.

Findings

Bijection between sequences and multitype forests via BFS

Extension of Lamperti transformation to multitype processes

Representation of processes through compound Poisson processes

Abstract

We obtain a bijection between some set of multidimensional sequences and this of $d$-type plane forests which is based on the breadth first search algorithm. This coding sequence is related to the sequence of population sizes indexed by the generations, through a Lamperti type transformation. The same transformation in then obtained in continuous time for multitype branching processes with discrete values. We show that any such process can be obtained from a $d^2$ dimensional compound Poisson process time changed by some integral functional. Our proof bears on the discretisation of branching forests with edge lengths.