  A Markov tree is a probabilistic graphical model for a random vector indexed
by the nodes of an undirected tree encoding conditional independence relations
between variables. One possible limit distribution of partial maxima of samples
from such a Markov tree is a max-stable H\"usler-Reiss distribution whose
parameter matrix inherits its structure from the tree, each edge contributing
one free dependence parameter. Our central assumption is that, upon marginal
standardization, the data-generating distribution is in the max-domain of
attraction of the said H\"usler-Reiss distribution, an assumption much weaker
than the one that data are generated according to a graphical model. Even if
some of the variables are unobservable (latent), we show that the underlying
model parameters are still identifiable if and only if every node corresponding
to a latent 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 when some variables are latent. A typical application is a
river network in the form of a tree where, on some locations, no data are
available. We illustrate the model and the identifiability criterion on a data
set of high water levels on the Seine, France, with two latent variables. The
structured H\"usler-Reiss distribution is found to fit the observed extremal
dependence patterns well. The parameters being identifiable we are able to
quantify tail dependence between locations for which there are no data.
