Multivariate Pareto distributions have been widely-used for modeling the simultaneous occurrence of extreme events over a high threshold. However, existing likelihood-based inference approaches are computationally prohibitive in high dimensions, due to the need to censor observations in the bulk of the distribution. In this work, we construct models for spatial extremes by exploiting the sparse conditional independence structure of multivariate Pareto distributions of H\”usler--Reiss type defined on trees. Such models have a simplified likelihood function, which can be computed more efficiently, thus opening the door to higher-dimensional extreme-value problems. We illustrate our methodology based on simulations, and if time allows by application to extremes from an environmental dataset.