  Comprehensive characterization of non-Poissonian, bursty temporal patterns
observed in various natural and social processes is crucial to understand 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 decomposing an event sequence into a set of IETs and a
burst tree, which exactly captures the structure of temporal correlations that
is entirely missing in the analysis of IET distributions. We apply the
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. 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 higher-order 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.
