  The Horton and Tokunaga branching laws provide a convenient framework for
studying self-similarity in random trees. The Horton self-similarity is a
weaker property that addresses the principal branching in a tree; it is a
counterpart of the power-law size distribution for elements of a branching
system. The stronger Tokunaga self-similarity addresses so-called side
branching. The Horton and Tokunaga self-similarity have been empirically
established in numerous observed and modeled systems, and proven for two
paradigmatic models: the critical Galton-Watson branching process with finite
progeny and the finite-tree representation of a regular Brownian excursion.
This study establishes the Tokunaga and Horton self-similarity for a tree
representation of a finite symmetric homogeneous Markov chain. We also extend
the concept of Horton and Tokunaga self-similarity to infinite trees and
establish self-similarity for an infinite-tree representation of a regular
Brownian motion. We conjecture that fractional Brownian motions are also
Tokunaga and Horton self-similar, with self-similarity parameters depending on
the Hurst exponent.
