Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue we develop a novel statistical method that describes extremal dependence taking advantage of the inherent tree-based dependence structure of the max-stable nested logistic distribution, and that identifies possible clusters of extreme variables using reversible jump Markov chain Monte Carlo techniques. Parsimonious representations are achieved when clusters of extreme variables are found to be completely independent. Moreover, we significantly decrease the computational complexity of full likelihood inference by deriving a recursive formula for the likelihood function of the nested logistic model. The method’s performance is verified through extensive simulation experiments which also compare different likelihood procedures. The new methodology is used to investigate the dependence relationships between extreme concentrations of multiple pollutants in California and how these concentrations are related to extreme weather conditions. Overall, we show that our approach allows for the representation of complex extremal dependence structures and has valid applications in multivariate data analysis, such as air pollution monitoring, where it can guide policymaking.