  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.
