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### Entropy rates for Horton self-similar trees

Evgenia V. Chunikhina
2018 Chaos
In this paper we examine planted binary plane trees. First, we provide an exact formula for the number of planted binary trees with given Horton-Strahler orders.  ...  Then, using the notion of entropy, we examine the structural complexity of random planted binary trees with N vertices.  ...  The formula for the number of planted binary trees with specific Horton-Strahler numbers is given in Section III.  ...

### Entropy rates for Horton self-similar trees [article]

Evgenia V. Chunikhina
2018 arXiv   pre-print
In this paper we examine planted binary plane trees. First, we provide an exact formula for the number of planted binary trees with given Horton-Strahler orders.  ...  Then, using the notion of entropy, we examine the structural complexity of random planted binary trees with N vertices.  ...  The formula for the number of planted binary trees with specific Horton-Strahler numbers is given in Section III.  ...

### Central limit theorem for the bifurcation ratio of a random binary tree

Ken Yamamoto, Yoshihiro Yamazaki
2009 Journal of Physics A: Mathematical and Theoretical
A topological structure of a binary tree is expressed by a binary sequence, and the Horton-Strahler indices are calculated by using the sequence.  ...  A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated.  ...  In the present paper, we propose a method of calculating Horton-Strahler indices of a binary tree by using binary sequence, and show numerical evidence for the validity of Eqs. (4).  ...

### A Brief History of Strahler Numbers [chapter]

Javier Esparza, Michael Luttenberger, Maximilian Schlund
2014 Lecture Notes in Computer Science
The Strahler number or Horton-Strahler number of a tree, originally introduced in geophysics, has a surprisingly rich theory.  ...  We sketch some milestones in its history, and its connection to arithmetic expressions, graph traversing, decision problems for context-free languages, Parikh's theorem, and Newton's procedure for approximating  ...  If you know of further work connected to the Strahler number, please contact us. Acknowledgments We thank Carlos Esparza for his help with some calculations.  ...

### The botanical beauty of random binary trees [chapter]

Luc Devroye, Paul Kruszewski
1996 Lecture Notes in Computer Science
We generate random binary trees by splitting based upon the beta distribution, and obtain the standard binary search trees as a special case.  ...  We present a simple mechanism for quickly rendering computer images of botanical trees based on random binary trees commonly found in computer science.  ...  However, since many families of random binary trees have logarithmic HORTON-STRAHLER numbers in the number of nodes (see e.g., DEVROYE AND KRUSZEWSKI (1994 ,1995 ), similar comparisons such as 10 , if  ...

### Page 7080 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews
; Montreal, QC) A note on the Horton-Strahler number for random binary search trees.  ...  P{S, > (1/log3+e) logn} =0, where S,, is the Horton-Strahler number of a random binary search tree with n nodes.  ...

### Universal features of dendrites through centripetal branch ordering

Alexandra Vormberg, Felix Effenberger, Julia Muellerleile, Hermann Cuntz, Boris S. Gutkin
2017 PLoS Computational Biology
We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific.  ...  Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain.  ...  Acknowledgments We are grateful to A. D. Bird and P. Jedlicka for helpful discussions.  ...

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

Yevgeniy Kovchegov, Ilya Zaliapin
2019 arXiv   pre-print
The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding.  ...  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.  ...  Finally, we thank the participants of the workshop Random Trees: Structure, Self-similarity, and Dynamics that took place during April -, , at the Centro de Investigación en Matemáticas (CIMAT), Guanajuato  ...

### Page 5544 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews
Jerrum (4-EDIN-C; Edinburgh) 95i:68059 68Q25 60C05 68M07 68R10 Devroye, Luc (3-MGL-C; Montreal, PQ); Kruszewski, Paul (3-MGL-C; Montreal, PQ) A note on the Horton-Strahler number for random trees.  ...  Summary: “We consider the Horton-Strahler number S,, for ran- dom equiprobable binary trees with n nodes.  ...

### Invariance and attraction properties of Galton-Watson trees [article]

Yevgeniy Kovchegov, Ilya Zaliapin
2020 arXiv   pre-print
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 {  ...  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.  ...  tail (see Discussion and Fig. ) and to Ed Waymire for multiple discussions about the topic of random self-similar trees.  ...

### Random self-similar trees: A mathematical theory of Horton laws

Yevgeniy Kovchegov, Ilya Zaliapin
2020 Probability Surveys
The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding.  ...  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.  ...  Acknowledgements First and foremost, we are grateful to Ed Waymire for his continuing advice, encouragement, and support on more levels than one.  ...

### Invariance and attraction properties of Galton–Watson trees

Yevgeniy Kovchegov, Ilya Zaliapin
2021 Bernoulli
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 {  ...  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 .  ...  tail (see Discussion and Figure 9 ) and to Ed Waymire for multiple discussions about the topic of random self-similar trees.  ...

### Critical Tokunaga model for river networks [article]

Yevgeniy Kovchegov, Ilya Zaliapin, Efi Foufoula-Georgiou
2021 arXiv   pre-print
Here we capitalize on a recently developed theory of random self-similar trees to introduce a one-parametric family of self-similar critical Tokunaga trees that elucidates the origin of Horton laws, Hack's  ...  The proposed family includes the celebrated Shreve's random topology model and extends to trees that approximate the observed river networks with realistic exponents.  ...  This makes the search for such a singularity much easier: one can only consider the restriction of the function f (z) on the real axis.  ...

### Random self-similar trees and a hierarchical branching process [article]

Yevgeniy Kovchegov, Ilya Zaliapin
2018 arXiv   pre-print
We study self-similarity in random binary rooted trees.  ...  In a well-understood case of Galton-Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts the tree leaves.  ...  YK wishes to thank Tom Kurtz for providing a feedback regarding infinite dimensional population  ...

### Tree Visualisation and Navigation Clues for Information Visualisation

Ivan Herman, Maylis Delest, Guy Melancon
1998 Computer graphics forum (Print)
This paper describes a visual clue for trees as well as an automatic folding (clustering) technique, both based on some mathematical concepts and results in combinatorics.  ...  Information visualisation very often requires good navigation aids on large trees, which represent the underlying abstract information.  ...  Part of the work has been financed by a grant of the Franco-Dutch research cooperation programme "Van Gogh".  ...
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