An interesting class of irregular algorithms is tree traversal algorithms, which repeatedly traverse various trees to perform efficient computations. Tree traversal algorithms form the algorithmic kernels in an important set of applications in scientific computing, computer graphics, bioinformatics, and data mining, etc. There has been increasing interest in understanding tree traversal algorithms, optimizing them, and applying them in a wide variety of settings. Crucially, while there are many possible optimizations for tree traversal algorithms, which optimizations apply to which algorithms is dependent on algorithmic characteristics. In this work, we present a suite of tree traversal kernels, drawn from diverse domains, called Treelogy, to explore the connection between tree traversal algorithms and state-of-the-art optimizations. We characterize these algorithms by developing an ontology based on their structural properties. The attributes extracted through our ontology, for a given traversal kernel, can aid in quick analysis of the suitability of platform- and application-specific as well as independent optimizations. We provide reference implementations of these kernels for three platforms: shared memory multicores, distributed memory systems, and GPUs, and evaluate their scalability.