For ease of exposition we describe a one dimensional system that is represented by a tree of depth D = 4, graphically depicted in Fig. . In this scheme the "leaf" nodes 16-31 represent a uniform division of the space containing the interacting particles, while nodes 1-15 on the lower levels of the tree are initially empty. We start by choosing the Ewald partitioning coefficient α such that the short-ranged kernel erfc αr12 r12 is below some threshold ϵ for all distances greater than the leaf width, R leaf :