  We develop comparison results for Markov tree distributions extending
ordering results from the literature on discrete time Markov processes and
recently studied ordering results for conditionally independent factor models
to tree structures. Based on fairly natural positive dependence conditions, our
main contribution is a comparison result with respect to the supermodular
order. Since this order is a pure dependence order, it has many applications in
optimal transport, finance, and insurance. As an illustrative example, we
consider hidden Markov models and study distributional robustness for
functionals of the random walk under model uncertainty. Further, we show that,
surprisingly, more general comparison results via the recently established
rearrangement-based Schur order for conditional distributions, which implies an
ordering of Chatterjee's rank correlation, do not carry over from star
structures to trees. Several examples and a detailed discussion of the
assumptions demonstrate the generality of our results and provide further
insights into the behavior of multidimensional distributions.
