Tree P{\'o}lya Splitting distributions for multivariate count data

In this article, we develop a new class of multivariate distributions adapted for count data, called Tree P{\'o}lya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependencies (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree P{\'o}lya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependency structures (covariance and correlations). The abundance of 17 species of Trichoptera recorded at 49 sites is used, on one hand, to illustrate the theoretical properties developed in this article on a concrete case, and on the other hand, to demonstrate the interest of this type of models, notably by comparing them to classical approaches in ecology or microbiome.