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We study self-similarity in random binary rooted trees. In a well-understoodcase of Galton-Watson trees, a distribution on a space of trees is said to beself-similar if it is invariant with respect to the operation of pruning, whichcuts the tree leaves. This only happens for the critical Galton-Watson tree (aconstant process progeny), which also exhibits other special symmetries. Weextend the prune-invariance setup to arbitrary binary trees with edge lengths.In this general case the class of self-similar processes becomes much richerand covers a variety of practically important situations. The main result isconstruction of the hierarchical branching processes that satisfy variousself-similarity definitions (including mean self-similarity and self-similarityin edge-lengths) depending on the process parameters. Taking the limit ofaveraged stochastic dynamics, as the number of trajectories increases, weobtain a deterministic system of differential equations that describes theprocess evolution. This system is used to establish a phase transition thatseparates fading and explosive behavior of the average process progeny. Wedescribe a class of critical Tokunaga processes that happen at the phasetransition boundary. They enjoy multiple additional symmetries and include thecelebrated critical binary Galton-Watson tree with independent exponential edgelength as a special case. Finally, we discuss a duality between trees andcontinuous functions, and introduce a class of extreme-invariant processes,constructed as the Harris paths of a self-similar hierarchical branchingprocess, whose local minima has the same (linearly scaled) distribution as theoriginal process.

Random self-similar trees and a hierarchical branching process