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We present a generic solution to a problem which lies at the heart of the unpredictable worst-case performance characteristics of a wide class of multi-dimensional index designs: those which employ a recursive partitioning of the data space. We then show how this solution can produce modified designs with fully predictable and controllable worst-case characteristics. In particular, we show how the recursive partitioning of an n-dimensional dataspace can be represented in such a way that the characteristics of the one-dimensional B-tree are preserved in n dimensions, as far as is topologically possible i.e. a representation guaranteeing logarithmic access and update time, while also guaranteeing a one-third minimum occupancy of both data and index nodes.