A note on the Horton-Strahler number for random trees

Devroye, Luc; Kruszewski, Paul

doi:10.1016/0020-0190(94)00135-9

Cited by 18 publications

(20 citation statements)

References 8 publications

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“…Following the pioneering work of Rodríguez‐Iturbe and Valdés [1979], they were incorporated into the theoretical scheme of the Geomorphological Instantaneous Unit Hydrograph (GIUH) and largely used to link the hydrological response to the geomorphological features [ Rodríguez‐Iturbe and Valdés , 1979; Gupta et al , 1980; Rosso , 1984; Gupta et al , 1986; Gupta and Mesa , 1988; Bras and Rodríguez‐Iturbe , 1989; Jin , 1992; Rinaldo et al , 1995; Rodríguez‐Iturbe and Rinaldo , 1997; Gupta and Waymire , 1998; Saco and Kumar , 2002a, 2002b, 2004; Bhunya et al , 2003, 2004, 2007, 2008; Rodriguez et al , 2005; Kumar et al , 2007; Singh et al , 2007; Lee et al , 2008], to estimate peak discharge using GIUH [ Sorman , 1995], to establish statistical scaling laws of mean annual discharge [ De Vries et al , 1994] and peak flows [ Gupta et al , 1996; Mantilla et al , 2006]. Horton‐Strahler's ordering scheme were also applied in other domains, such as coding binary trees [ Devroye and Kruszewski , 1994; Kruszewski , 1999; Zaliapin et al , 2006], or establishing hierarchical structures of cities [ Chen and Zhou , 2008].…”

Section: Introductionmentioning

confidence: 99%

Definition of new equivalent indices of Horton‐Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph

Moussa

2009

Water Resources Research

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[1] The Horton-Strahler concept has been important in river basin geomorphology; it describes scaling properties and identifies GIUH but has also attracted criticisms, because it depends on the threshold area S, which is used to extract the channel network from digital elevation models (DEM), on the position of the outlet and on the few number of ratios used to identify the scaling laws. To overcome these limitations, this paper proposes new indices independently of S and which have similar properties than the Horton-Strahler ratios. Applications were conducted on seven French basins. The methodology consists on analyzing the morphometric properties, such as the drained area, the number of sources, and the length of the channel network as a function of S. The results show that Horton-Strahler ratios vary considerably with S. Particularly, while the W-order length L W is a staircase function of S, results show that the channel network is articulated around two main nodes independent of S, which are descriptors of L W . Then, we compared the properties of Horton-Strahler laws to those obtained from self-similarity analysis, which enables to define new indices independent of S. These indices are calculated automatically from DEM, have similar properties as the Horton-Strahler ratios, and are used to calculate the GIUH independently of S. The four new descriptors R Be , R Le , R Ae , and L e , which have similar properties as R B , R L , R A , and L W respectively, are equivalent to those of Horton-Strahler's laws and can be used as indicators of hydrological similarity for catchment comparison and regionalization.

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Section: Introductionmentioning

confidence: 99%

Definition of new equivalent indices of Horton‐Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph

Moussa

2009

Water Resources Research

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“…Later, Flajolet and Prodinger extend the analysis to trees with both binary and unary inner nodes [17]. Finally, Devroye and Kruszewski show in [5] that the probability that the Strahler number of a random binary tree with n nodes deviates by at least k from the expected Strahler number of log 4 n is bounded from above by 2 4 k , that is, the Strahler number is highly concentrated around its expected value.…”

Section: Distribution Of Strahler Numbersmentioning

confidence: 99%

A Brief History of Strahler Numbers

Esparza

Luttenberger

Schlund

2014

Language and Automata Theory and Applications

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“…• Step VI. Taking ψ (N ) (t) ≡ η (N ) (t), we compare the difference equation (5) with (6), and bound the error |η (N ) (t) − η(t)| for all t ∈ [0, K]. Specifically, we show that with probability greater than P…”

Section: Hydrodynamic Limitmentioning

confidence: 99%

Horton self-similarity of Kingman’s coalescent tree

Kovchegov¹,

Zaliapin²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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The paper establishes Horton self-similarity for a tree representation of Kingman's coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations that describes evolution of the number of branches of a given Horton-Strahler order in a tree that represents Kingman's N -coalescent, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman's tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.This study is a first step toward exploring Horton self-similarity with ratio R = 2, 4. We consider here the tree generated by Kingman's coalescent process with N particles. The main result is a weaker form of Horton self-similarity, called here root-Horton law. The Horton ratio is estimated numerically as R = 3.043827 . . .. We also establish a close relation between the combinatorial tree representations of Kingman's N -coalescent and a combinatorial level set tree for a sequence of i.i.d. random variables (referred to as discrete white noise), which implies Horton self-similarity for the latter. These findings add two important classes of processes -Kingman's coalescent and discrete white noise -to the realm of Horton selfsimilar systems.The paper is organized as follows. Section 2 describes Horton-Strahler ordering of tree branches and the related concept of Horton self-similarity. Kingman's coalescent process and its tree representation are defined in Sect. 3. The main results are summarized in Sect. 4. Section 5 introduces the Smoluchowski-Horton system of equations that describes the dynamics of Horton-Strahler branches in Kingman's coalescent. This section also establishes the validity of the Smoluchowski-Horton equations, as well as the existence of some related quantities, in hydrodynamic limit. A proof of the existence of root-Horton law for Kingman's coalescent is presented in Sect. 6. Section 7 demonstrates a connection between the combinatorial tree representation of Kingman's N -coalescent process and combinatorial level set tree of a discrete white noise. The Smoluchowski-Horton system for a general coalescent process with collision kernel is written in Sect. 8. Section 9 concludes. Self-similar treesThis section defines Horton self-similarity for rooted binary trees. Rooted trees.A graph G = (V, E) is a collection of vertices V = {v i }, 1 ≤ i ≤ N V and edges E = {e k }, 1 ≤ k ≤ N E . In a simple undirected graph each edge is defined as an unordered pair of distinct vertices: ∀ 1 ≤ k ≤ N E , ∃! 1 ≤ i, j ≤ N V , i = j such that e k = (v i , v j ) and we say that the edge k connects vertices v i and v j . Furthermore, each pair of vertices in a simple graph may have at most one connecting edge. A tree is a connected simple graph T = (V, E) without cycles. In a rooted tree, one node is designated as a root; this imposes a natural direction of edges as well as the parent-child relationship between the vertices. Specifically, of the two connected vertices the one closest to th...

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A note on the Horton-Strahler number for random trees

Cited by 18 publications

References 8 publications

Definition of new equivalent indices of Horton‐Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph

Definition of new equivalent indices of Horton‐Strahler ratios for the derivation of the Geomorphological Instantaneous Unit Hydrograph

A Brief History of Strahler Numbers

Horton self-similarity of Kingman’s coalescent tree

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