#### Yoder, M. R.,Van Aalsburg, J.,Turcotte, D. L.,Abaimov, S. G.,Rundle, J. B.

Aftershock statistics provide a wealth of data that can be used to better understand earthquake physics. Aftershocks satisfy scale-invariant Gutenberg-Richter (GR) frequency-magnitude statistics. They also satisfy Omori's law for power-law seismicity rate decay and BAyenth's law for maximum-magnitude scaling. The branching aftershock sequence (BASS) model, which is the scale-invariant limit of the epidemic-type aftershock sequence model (ETAS), uses these scaling laws to generate synthetic aftershock sequences. One objective of this paper is to show that the branching process in these models satisfies Tokunaga branching statistics. Tokunaga branching statistics were originally developed for drainage networks and have been subsequently shown to be valid in many other applications associated with complex phenomena. Specifically, these are characteristic of a universality class in statistical physics associated with diffusion-limited aggregation. We first present a deterministi! c version of the BASS model and show that it satisfies the Tokunaga side-branching statistics. We then show that a fully stochastic BASS simulation gives similar results. We also study foreshock statistics using our BASS simulations. We show that the frequency-magnitude statistics in BASS simulations scale as the exponential of the magnitude difference between the mainshock and the foreshock, inverse GR scaling. We also show that the rate of foreshock occurrence in BASS simulations decays inversely with the time difference between foreshock and mainshock, an inverse Omori scaling. Both inverse scaling laws have been previously introduced empirically to explain observed foreshock statistics. Observations have demonstrated both of these scaling relations to be valid, consistent with our simulations. ETAS simulations, in general, do not generate BAyenth's law and do not generate inverse GR scaling.