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