The Horton-Strahler Number of Conditioned Galton-Watson Trees
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| Format: | Preprint |
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2020
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| _version_ | 1866915318552592384 |
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| author | Brandenberger, Anna M. Devroye, Luc Reddad, Tommy |
| author_facet | Brandenberger, Anna M. Devroye, Luc Reddad, Tommy |
| contents | The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_08613 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Horton-Strahler Number of Conditioned Galton-Watson Trees Brandenberger, Anna M. Devroye, Luc Reddad, Tommy Probability Combinatorics 60C05, 60J80 (Primary) 05C80, 05C05 (Secondary) The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case. |
| title | The Horton-Strahler Number of Conditioned Galton-Watson Trees |
| topic | Probability Combinatorics 60C05, 60J80 (Primary) 05C80, 05C05 (Secondary) |
| url | https://arxiv.org/abs/2010.08613 |