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Entropy rates for Horton self-similar trees
2018
Chaos
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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. ...
doi:10.1063/1.5048965
pmid:30180610
fatcat:okfrkqczkvdadp7v5wtbme3cge
Entropy rates for Horton self-similar trees
[article]
2018
arXiv
pre-print
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2018 pre-print arXiv:1804.06989v1 fatcat:vk5myjuff5b4dct25dhudp7u4e
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. ...
arXiv:1804.06989v2
fatcat:tzzg43oq5vbafewa6nzciag5ri
Central limit theorem for the bifurcation ratio of a random binary tree
2009
Journal of Physics A: Mathematical and Theoretical
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2009 pre-print arXiv:0904.2043v1 fatcat:twg3mbiqh5hmnjcgfsfbvlhn7aOther Versions
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). ...
doi:10.1088/1751-8113/42/41/415002
fatcat:okyxyrrcxzbwbbnbnanh2rezvq
A Brief History of Strahler Numbers
[chapter]
2014
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-319-04921-2_1
fatcat:dgzdfkrenjhlpjmatpxbbhf4t4
The botanical beauty of random binary trees
[chapter]
1996
Lecture Notes in Computer Science
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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 ...
doi:10.1007/bfb0021801
fatcat:pj3sycivvbgjtjdamzo6nnzjza
Page 7080 of Mathematical Reviews Vol. , Issue 99j
[page]
1999
Mathematical Reviews
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; 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
2017
PLoS Computational Biology
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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. ...
doi:10.1371/journal.pcbi.1005615
pmid:28671947
pmcid:PMC5515450
fatcat:alpmzor5iremdknm6t3cfp3cca
Random Self-Similar Trees: A mathematical theory of Horton laws
[article]
2019
arXiv
pre-print
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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 ...
arXiv:1905.02629v2
fatcat:zxm5e2pju5dm5fvejm65323xr4
Page 5544 of Mathematical Reviews Vol. , Issue 95i
[page]
1995
Mathematical Reviews
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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]
2020
arXiv
pre-print
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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. ...
arXiv:1911.08095v3
fatcat:yaqersagxbcfdinr65ewwrfzfi
Random self-similar trees: A mathematical theory of Horton laws
2020
Probability Surveys
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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. ...
doi:10.1214/19-ps331
fatcat:yy333bgrlzcp7lvdtcqzttg5sq
Invariance and attraction properties of Galton–Watson trees
2021
Bernoulli
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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. ...
doi:10.3150/20-bej1292
fatcat:j7fw3wvdivhptpx4et5uogvokm
Critical Tokunaga model for river networks
[article]
2021
arXiv
pre-print
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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. ...
arXiv:2106.02991v1
fatcat:stkfbla3wrf5nkbxz5sqev7mwq
Random self-similar trees and a hierarchical branching process
[article]
2018
arXiv
pre-print
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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 ...
arXiv:1608.05032v4
fatcat:mdehvanl6fc77bmjlsvm3tbho4
Tree Visualisation and Navigation Clues for Information Visualisation
1998
Computer graphics forum (Print)
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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". ...
doi:10.1111/1467-8659.00235
fatcat:e3bxxzhjtvasfi7hb52gvzkwxu
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