Two cases of multinomial distributions can be considered: the singular -- for modeling univariate categorical data -- and the non-singular -- for modeling multivariate count data. Considering this latter case, we introduce sum-compound multinomial distributions that encompass re-parameterization of non-singular multinomial and negative multinomial distributions. These compound distributions also enable to generalize univariate distributions and their maximum likelihood estimators to the multivariate case. These distributions are used to address the inference of discrete-state models for tree-structured data. In particular, they allow to introduce parametric multi-type branching processes that can be easily interpreted and efficiently estimated on the basis of data of limited size. The proposed modeling approach is illustrated using plant architecture data sets.