While our proposed algorithm for SGT is strikingly similar to that used in FMM, there are some notable differences. The upwards pass, in which we gather vectors at lower levels of the tree is analogous to the coarsegraining of the multipoles at the same stage of the FMM. However, in FMM a downward pass is required, in which the centers of the Taylor series expansion is translated to the center of each interacting box. While asymptotically linear scaling in effort, this process involves spherical harmonic translation operators and has a relatively high computational prefactor. In the possible SGT schemes that we sketch out above, the downward pass only involves additions and subtractions, either through a recursive scheme to propagate the potentials down the tree or, more crudely, by creating an explicit list of remote nodes for each leaf node. Another crucial distinction between FMM and SGT is that the latter makes no appeal to the concept of well-separatedness: the factorization shown in eqn. ( ) is only valid for non-overlapping densities, while eqn. ( ) makes no such assumption.