Three fascinating concepts, which seem at first to be entirely unrelated to each other, are in fact in one-to-one correspondence, via three beautiful algorithms. One of them is a data structure involving a pair of binary trees, introduced by Serge Dulucq and Olivier Guibert in 1998; another is a class of permutations introduced by Glen Baxter in 1964; and the third is a geometric understanding of the decomposition of rectangles into rectangles, introduced by Hiroshi Murata, Kunihiro Fujiyoshi, Tomomi Watanabe, and Yoji Kajitani in 1997.

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