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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 inhierarchical branching systems, the Horton laws have been rediscovered in manydisciplines ranging from geomorphology to genetics to computer science.Attempts to build a mathematical foundation behind the Horton laws during the1990s revealed their close connection to the operation of pruning -- erasing atree from the leaves down to the root. This survey synthesizes recent resultson invariances and self-similarities of tree measures under various forms ofpruning. We argue that pruning is an indispensable instrument for describingbranching structures and representing a variety of coalescent and annihilationdynamics. The Horton laws appear as a characteristic imprint ofself-similarity, which settles some questions prompted by geophysical data.

Random Self-Similar Trees: A mathematical theory of Horton laws