A note on the Horton-Strahler number for random trees

Devroye, Luc; Kruszewski, Paul

doi:10.1016/0020-0190(95)00114-r

Cited by 17 publications

(9 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Horton-Strahler orders can be equivalently defined via hierarchical counting [3,9,[19][20][21]. The first such definition beyond the binary case appeared in [2].…”

Section: (Initial and Terminal Vertex Of A Branch)mentioning

confidence: 99%

Invariance and attraction properties of Galton–Watson trees

Kovchegov

Zaliapin

2021

Bernoulli

View full text Add to dashboard Cite

We give a description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton-Watson tree and a one-parameter family of critical Galton-Watson trees with offspring distribution {q k } that has a power tail q k ∼ Ck −(1+1/q 0 ) , where q 0 ∈ (1/2, 1). Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the tail behavior of the initial Galton-Watson offspring distribution. The invariant measures satisfy the Toeplitz property for the Tokunaga coefficients and obey the Horton law with exponent R = (1 − q 0 ) −1/q 0 .

show abstract

“…The Horton-Strahler orders can be equivalently defined via hierarchical counting [3,9,[19][20][21]. The first such definition beyond the binary case appeared in [2].…”

Section: (Initial and Terminal Vertex Of A Branch)mentioning

confidence: 99%

Invariance and attraction properties of Galton–Watson trees

Kovchegov

Zaliapin

2021

Bernoulli

View full text Add to dashboard Cite

show abstract

“…Two topological characteristics are used: the Horton-Strahler number (HN) of a node and the centrifugal order of a segment. The centrifugal order (CO) of a segment denotes its topological distance to the root segment [12], whereas the HN of a node or a tree is a numerical measure of its branching complexity [13]. The HN of each graph is here normalized with respect to its maximum.…”

Section: Improved Gaussian Process Graph Matchingmentioning

confidence: 99%

Biomechanics-based graph matching for augmented CT-CBCT

2018

View full text Add to dashboard Cite

Purpose: Augmenting intraoperative cone beam computed tomography (CBCT) images with preoperative computed tomography (CT) data in the context of imageguided liver therapy is proposed. The expected benefit is an improved visualization of tumor(s), vascular system and other internal structures of interest. Method: An automatic elastic registration based on matching of vascular trees extracted from both the preoperative and intraoperative images is presented. Although methods dedicated to non-rigid graph matching exist, they are not efficient when large intraoperative deformations of tissues occur, as is the case during the liver surgery. The contribution is an extension of the graph matching algorithm using Gaussian process regression (GPR) [1]: First, an improved GPR matching is introduced by imposing additional constraints during the matching when the number of hypothesis is large; like the original algorithm, this extended version does not require a manual initialization of matching. Second, a fast biomechanical model is employed to make the method capable of handling large deformations. Results: The proposed automatic intraoperative augmentation is evaluated on both synthetic and real data. It is demonstrated that the algorithm is capable of handling large deformations, thus being more robust and reliable than previous approaches. Moreover, the time required to perform the elastic registration is compatible with the intraoperative navigation scenario. Conclusion: A biomechanics-based graph matching method, which can handle large deformations and augment intraoperative CBCT, is presented and evaluated.

show abstract

“…Other results on the Horton-Strahler analysis and tree structures are found in Refs. [24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Topological Self-similarity on the Random Binary-Tree Model

Yamamoto

Yamazaki

2010

J Stat Phys

View full text Add to dashboard Cite

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological selfsimilarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented.

show abstract

A note on the Horton-Strahler number for random trees

Cited by 17 publications

References 7 publications

Invariance and attraction properties of Galton–Watson trees

Invariance and attraction properties of Galton–Watson trees

Biomechanics-based graph matching for augmented CT-CBCT

Topological Self-similarity on the Random Binary-Tree Model

Contact Info

Product

Resources

About