...#stack>).
These ``magic'' sizes are actually the result of numerous constraints; for a complete description of the constraints which lead to these configurations, see [DeH90].

...root.
[Lei85] and [GL85] do not necessarily prove universality of fat-trees with the generality used here. Given the structure of their proofs, it is likely that their work can be generalized in this manner.

...cube.
As this progression continues, it becomes necessary to take the size of the ``sides'' of each square into account, making the increase slightly more than a factor of four. The additional size, though, is only a second order effect.

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