The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested here. We define a geometric branching processes G(s) that generates self-similar rooted trees. We establish the equivalence between the invariance of G(s) with respect to a time shift and a one-parametric version of the Tokunaga condition. In the parameter region where the process satisfies the Tokunaga condition (and hence is time invariant), G(s) enjoys many of the symmetries observed in a critical binary Galton-Watson branching process and reproduce the latter for a particular parameter value.
This is a joint work with Ilya Zaliapin (University of Nevada, Reno).