Abstract
The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding. Ubiquitous in hierarchical branching systems, the Horton laws have been rediscovered in many disciplines ranging from geomorphology to genetics to computer science. Attempts to build a mathematical foundation behind the Horton laws during the 1990s revealed their close connection to the operation of pruning – erasing a tree from the leaves down to the root. This survey synthesizes recent results on invariances and self-similarities of tree measures under various forms of pruning. We argue that pruning is an indispensable instrument for describing branching structures and representing a variety of coalescent and annihilation dynamics. The Horton laws appear as a characteristic imprint of self-similarity, which settles some questions prompted by geophysical data.
Highlights
Invariance of the critical binary Galton-Watson tree measure with respect to pruning that begins at the leaves and progresses down to the tree root has been recognized since the late 1980s
We show (Cor. 21 of Thm. 28) that the shock tree is isometric to the level set tree of the initial potential, and the model evolution is equivalent to a generalized dynamical pruning of the shock tree, with the pruning function equal to the total tree length (Thm. 30)
We introduce here a class of hierarchical branching processes that enjoy all of the symmetries discussed in this work – Horton self-similarity, criticality, timeinvariance, strong Horton law, Tokunaga self-similarity, and have independently distributed edge lengths
Summary
Invariance of the critical binary Galton-Watson tree measure with respect to pruning (erasure) that begins at the leaves and progresses down to the tree root has been recognized since the late 1980s. Both continuous [123] and discrete [34] versions of prunings have been studied. The prune-invariance of the trees naturally translates to the symmetries of the respective Harris paths [80] The richness of such a connection is supported by the well-studied embeddings of the Galton-Watson trees in the excursions of random walks and Brownian motions We keep references to a minimum, and indicate survey sections where one can find future information
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