A Markov tree is a probabilistic graphical model for a random vector by which conditional independence relations between variables are encoded via an undirected tree and each node corresponds to a variable. One possible max-stable attractor for such a model is a Hüsler–Reiss extreme value distribution whose variogram matrix inherits its structure from the tree, each edge contributing one free dependence parameter. Even if some of the variables are latent, as can occur on junctions or splits in a river network, the underlying model parameters are still identifiable if and only if every node corresponding to a missing variable has degree at least three. Three estimation procedures, based on the method of moments, maximum composite likelihood, and pairwise extremal coefficients, are proposed for usage on multivariate peaks over thresholds data. The model and the methods are illustrated on a dataset of high water levels at several locations on the Seine network. The structured Hüsler–Reiss distribution is found to fit the observed extremal dependence well, and the fitted model confirms the importance of flow-connectedness for the strength of dependence between high water levels, even for locations at large distance apart.