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We construct a tree-based dependence structure for the representation ofbinomial, Poisson and Gaussian random vectors having a given covariance matrix,using sums of independent random variables. This construction allows us tocharacterize the supermodular ordering of such random vectors via thecomponentwise ordering of their covariance matrices. Our method relies on therepresentation of dependent components using binary trees on the discrete$d$-dimensional hypercube $C_d$, and on M\"obius inversion techniques. In thecase of Poisson random vectors this approach involves L\'evy measures on $C_d$,and it is consistent with the approximation of Poisson and multivariateGaussian random vectors by binomial vectors.

Supermodular ordering of binomial, Poisson and Gaussian random vectors by tree-based correlations