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We propose a comprehensive survey of tree kernels through the lens of the mapping kernels framework. We argue that most existing tree kernels, as well as many more that are presented for the first time in this paper, fall into a typology of kernels whose seemingly intricate computation can be efficiently factorized to yield polynomial time algorithms. Despite this fact, we argue that a naive implementation of such kernels remains prohibitively expensive to compute. We propose an approach whereby some computations for smaller trees are cached, which speeds up considerably the computation of all these tree kernels. We provide experimental evidence of this fact as well as preliminary results on the performance of these kernels.

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