The hierarchical organization and emergence of scaling laws in complex systems—geophysical, biological, technological, and socioeconomic—have been the topic of extensive research at the turn of the twentieth century. Although significant progress has been achieved, the mathematical origin of and relation among scaling laws for different system attributes remain unsettled. Paradigmatic examples are the Gutenberg–Richter law of seismology and Horton’s laws of geomorphology. We review the results that clarify the appearance, parameterization, and implications of scaling laws in hierarchical systems conceptualized by tree graphs. A recently formulated theory of random self-similar trees yields a suite of results on scaling laws for branch attributes, tree fractal dimension, power-law distributions of link attributes, and power-law relations between distinct attributes. Given the relevance of power laws to extreme events and hazards, our review informs related theoretical and modeling efforts and provides a framework for unified analysis in hierarchical complex systems.

复杂系统（地球物理、生物、技术和社会经济）中的等级组织和尺度定律的出现一直是 20 世纪之交广泛研究的主题。尽管已经取得了重大进展，但不同系统属性的标度律的数学起源和关系仍未确定。典型的例子是 Gutenberg-Richter 地震学定律和 Horton 地貌学定律。我们回顾了结果，这些结果阐明了由树图概念化的层次系统中的缩放定律的外观、参数化和含义。最近制定的随机自相似树理论产生了一系列关于分支属性的缩放定律、树分形维数、链接属性的幂律分布的结果，以及不同属性之间的幂律关系。鉴于幂律与极端事件和危险的相关性，我们的评论为相关的理论和建模工作提供了信息，并为分层复杂系统的统一分析提供了框架。