Invariant Galton–Watson trees: metric properties and attraction with respect to generalized dynamical pruning

The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

Critical Tokunaga model for river networks

The hierarchical organization and self-similarity in river basins have been topics of extensive research in hydrology and geomorphology starting with the pioneering work of Horton in 1945. Despite significant theoretical and applied advances, however, the mathematical origin of and relation among Horton's laws for different stream attributes remain unsettled. Here we capitalize on a recently developed theory of random self-similar trees to elucidate the origin of Horton's laws, Hack's laws, basin fractal dimensions, power-law distributions of link attributes, and power-law relations between distinct attributes. In particular, we introduce a one-parametric family of self-similar critical Tokunaga trees that includes the celebrated Shreve's random topology model and extends to trees that approximate the observed river networks with realistic exponents. The results offer tools to increase our understanding of landscape organization under different hydroclimatic forcings, and to extend scaling relationships useful for hydrologic prediction to resolutions higher than those observed.