Understanding the effects of the choice of the tree on the joint distribution of a tree-structured Markov random field (MRF) is crucial for fully exploiting the intelligibility of such probabilistic graphical models. Tools must be developed in this regard: this is the overarching objective of this paper. Our discussion is two-fold. First, we examine concepts specific to network centrality theory. We put forth a new conception of centrality for MRFs that not only accounts for the tree topology, but also the underlying stochastic dynamics. In this vein, we compare synecdochic pairs, random vectors comprising a MRF's component and its sum, using stochastic orders. The resulting orderings are transferred to risk-allocation quantities, which therefore serve as new centrality indices tailored to our stochastic framework. Second, we shed light on the influence of the tree's shapes, by establishing convex orderings for MRFs encrypted on trees of different shapes. This results in the design of a new partial order on tree shapes. This analysis is done within the framework of a propagation-based family of tree-structured MRFs with the uncommon property of having fixed Poisson marginal distributions unaffected by the dependence scheme. This work is a first step into the analysis of MRFs' trees' shapes and a stepping stone to extending this analysis to a broader framework.