  Multivariate extreme value distributions are a common choice for modelling
multivariate extremes. In high dimensions, however, the construction of
flexible and parsimonious models is challenging. We propose to combine
bivariate extreme value distributions into a Markov random field with respect
to a tree. Although in general not an extreme value distribution itself, this
Markov tree is attracted by a multivariate extreme value distribution. The
latter serves as a tree-based approximation to an unknown extreme value
distribution with the given bivariate distributions as margins. Given data, we
learn an appropriate tree structure by Prim's algorithm with estimated pairwise
upper tail dependence coefficients or Kendall's tau values as edge weights. The
distributions of pairs of connected variables can be fitted in various ways.
The resulting tree-structured extreme value distribution allows for inference
on rare event probabilities, as illustrated on river discharge data from the
upper Danube basin.
