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Entropy rates for Horton self-similar trees

2018
*
Chaos
*

In this paper we examine planted

doi:10.1063/1.5048965
pmid:30180610
fatcat:okfrkqczkvdadp7v5wtbme3cge
*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]

2018
*
arXiv
*
pre-print

In this paper we examine planted

arXiv:1804.06989v2
fatcat:tzzg43oq5vbafewa6nzciag5ri
*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

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]

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]

1996
*
Lecture Notes in Computer Science
*

We generate

doi:10.1007/bfb0021801
fatcat:pj3sycivvbgjtjdamzo6nnzjza
*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

2017
*
PLoS Computational Biology
*

We report

doi:10.1371/journal.pcbi.1005615
pmid:28671947
pmcid:PMC5515450
fatcat:alpmzor5iremdknm6t3cfp3cca
*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]

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]

2020
*
arXiv
*
pre-print

Under

arXiv:1911.08095v3
fatcat:yaqersagxbcfdinr65ewwrfzfi
*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

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

2021
*
Bernoulli
*

Under

doi:10.3150/20-bej1292
fatcat:j7fw3wvdivhptpx4et5uogvokm
*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]

2021
*
arXiv
*
pre-print

Here we capitalize

arXiv:2106.02991v1
fatcat:stkfbla3wrf5nkbxz5sqev7mwq
*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]

2018
*
arXiv
*
pre-print

We study self-similarity in

arXiv:1608.05032v4
fatcat:mdehvanl6fc77bmjlsvm3tbho4
*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

1998
*
Computer graphics forum (Print)
*

This paper describes

doi:10.1111/1467-8659.00235
fatcat:e3bxxzhjtvasfi7hb52gvzkwxu
*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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