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 Probability Surveys > Vol. 17 (2020) open journal systems 


Random self-similar trees: A mathematical theory of Horton laws

Yevgeniy Kovchegov, Oregon State University
Ilya Zaliapin, University of Nevada Reno


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.

AMS 2000 subject classifications: Primary 05C05, 05C80; secondary 05C63, 58-02

Keywords: Random self-similar trees; Horton laws

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Kovchegov, Yevgeniy, Zaliapin, Ilya, Random self-similar trees: A mathematical theory of Horton laws, Probability Surveys, 17, (2020), 1-213 (electronic). DOI: 10.1214/19-PS331.

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