# On the Horton-Strahler Number for Combinatorial Tries

• Volume: 34, Issue: 4, page 279-296
• ISSN: 0988-3754

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## Abstract

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In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is asymptotically given by $\frac{{4}^{2\beta -\alpha }log\left(\alpha \right)\left(2\beta -1\right)\left(\alpha +1\right)\left(\alpha +2\right)\left(\genfrac{}{}{0pt}{}{2\alpha +1}{\alpha -1}\right)}{8\sqrt{\pi }{\alpha }^{3/2}log\left(2\right)\left(\beta -1\right)\beta {\left(\genfrac{}{}{0pt}{}{2\beta }{\beta }\right)}^{2}}$ provided that α and β grow in some fixed proportion ρ when α → ∞ . A similar result is shown for trees with α internal nodes but with an arbitrary number of keys.

## How to cite

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Nebel, Markus E.. "On the Horton-Strahler Number for Combinatorial Tries." RAIRO - Theoretical Informatics and Applications 34.4 (2010): 279-296. <http://eudml.org/doc/222082>.

@article{Nebel2010,
abstract = { In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is asymptotically given by $$\frac\{4^\{2\beta-\alpha\}\log(\alpha)(2\beta-1)(\alpha+1)(\alpha+2)\{2\alpha+1\choose \alpha-1\}\}\{8\sqrt\{\pi\}\alpha^\{3/2\}\log(2)(\beta-1)\beta\{2\beta\choose \beta\}^2\}$$ provided that α and β grow in some fixed proportion ρ when α → ∞ . A similar result is shown for trees with α internal nodes but with an arbitrary number of keys. },
author = {Nebel, Markus E.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Horton-Strahler number; binary tries; binary trees},
language = {eng},
month = {3},
number = {4},
pages = {279-296},
publisher = {EDP Sciences},
title = {On the Horton-Strahler Number for Combinatorial Tries},
url = {http://eudml.org/doc/222082},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Nebel, Markus E.
TI - On the Horton-Strahler Number for Combinatorial Tries
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 4
SP - 279
EP - 296
AB - In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is asymptotically given by $$\frac{4^{2\beta-\alpha}\log(\alpha)(2\beta-1)(\alpha+1)(\alpha+2){2\alpha+1\choose \alpha-1}}{8\sqrt{\pi}\alpha^{3/2}\log(2)(\beta-1)\beta{2\beta\choose \beta}^2}$$ provided that α and β grow in some fixed proportion ρ when α → ∞ . A similar result is shown for trees with α internal nodes but with an arbitrary number of keys.
LA - eng
KW - Horton-Strahler number; binary tries; binary trees
UR - http://eudml.org/doc/222082
ER -

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