We assess advantages of expressing tree-structured Ising models via their mean parameterization rather than their commonly chosen canonical parameterization. This includes fixedness of marginal distributions, often convenient for dependence modeling, and the dispelling of the intractable normalizing constant otherwise hindering Ising models. We derive an analytic expression for the joint probability generating function of mean-parameterized tree-structured Ising models, conferring efficient computation methods for the distribution of the sum of its constituent random variables. The mean parameterization also allows for a stochastic representation of Ising models, providing straightforward sampling methods. We furthermore show that Markov random fields with fixed Poisson marginal distributions may act as an efficient and accurate approximation for tree-structured Ising models, in the spirit of Poisson approximation.