  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 in this work. We
define a geometric branching processes $\mathcal{G}(s)$ that generates
self-similar rooted trees. The main result establishes the equivalence between
the invariance of $\mathcal{G}(s)$ 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),
$\mathcal{G}(s)$ enjoys many of the symmetries observed in a critical binary
Galton-Watson branching process and reproduce the latter for a particular
parameter value.
