Comprehensive characterization of non-Poissonian, bursty temporal patterns observed in various natural and social processes is crucial for understanding the underlying mechanisms behind such temporal patterns. Among them bursty event sequences have been studied mostly in terms of interevent times (IETs), while the higher-order correlation structure between IETs has gained very little attention due to the lack of a proper characterization method. In this paper we propose a method of representing an event sequence by a burst tree, which is then decomposed into a set of IETs and an ordinal burst tree. The ordinal burst tree exactly captures the structure of temporal correlations that is entirely missing in the analysis of IET distributions. We apply this burst-tree decomposition method to various datasets and analyze the structure of the revealed burst trees. In particular, we observe that event sequences show similar burst-tree structure, such as heavy-tailed burst-size distributions, despite of very different IET distributions. This clearly shows that the IET distributions and the burst-tree structures can be separable. The burst trees allow us to directly characterize the preferential and assortative mixing structure of bursts responsible for the higher-order temporal correlations. We also show how to use the decomposition method for the systematic investigation of such correlations captured by the burst trees in the framework of randomized reference models. Finally, we devise a simple kernel-based model for generating event sequences showing appropriate higher-order temporal correlations. Our method is a tool to make the otherwise overwhelming analysis of higher-order correlations in bursty time series tractable by turning it into the analysis of a tree structure.