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

## TL;DR

This paper develops a mathematical framework for understanding Horton laws in hierarchical trees, linking their self-similarity and invariance properties to pruning operations, with applications across various scientific disciplines.

## Contribution

It provides a unified mathematical theory connecting Horton laws to pruning and self-similarity in trees, advancing the understanding of branching structures.

## Key findings

- Horton laws are characterized as invariants under pruning.
- Self-similarity explains the universality of Horton laws across disciplines.
- Pruning operations are essential for modeling branching and coalescent processes.

## 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.