### Abstract

In order to formulate and examine the central limit theorem for a binary tree numerically, a method for generating random binary trees is presented. We first propose the correspondence between binary trees and a certain type of binary sequences (which we call Dyck sequences). Then, the method for generating random Dyck sequences is shown. Also, we propose the method of branch ordering of a binary tree by means of only the corresponding Dyck sequence. We confirm that the method is in good consistency with the topological analysis of binary trees known as the Horton-Strahler analysis. Two types of central limit theorem are numerically confirmed, and the obtained results are expressed in simple forms. Furthermore, the proposed method is available for a wide range of the topological analysis of binary trees.

Original language | English |
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Article number | 415002 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 42 |

Issue number | 41 |

DOIs | |

Publication status | Published - 2009 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

**Central limit theorem for the bifurcation ratio of a random binary tree.** / Yamamoto, Ken; Yamazaki, Yoshihiro.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 42, no. 41, 415002. https://doi.org/10.1088/1751-8113/42/41/415002

}

TY - JOUR

T1 - Central limit theorem for the bifurcation ratio of a random binary tree

AU - Yamamoto, Ken

AU - Yamazaki, Yoshihiro

PY - 2009

Y1 - 2009

N2 - In order to formulate and examine the central limit theorem for a binary tree numerically, a method for generating random binary trees is presented. We first propose the correspondence between binary trees and a certain type of binary sequences (which we call Dyck sequences). Then, the method for generating random Dyck sequences is shown. Also, we propose the method of branch ordering of a binary tree by means of only the corresponding Dyck sequence. We confirm that the method is in good consistency with the topological analysis of binary trees known as the Horton-Strahler analysis. Two types of central limit theorem are numerically confirmed, and the obtained results are expressed in simple forms. Furthermore, the proposed method is available for a wide range of the topological analysis of binary trees.

AB - In order to formulate and examine the central limit theorem for a binary tree numerically, a method for generating random binary trees is presented. We first propose the correspondence between binary trees and a certain type of binary sequences (which we call Dyck sequences). Then, the method for generating random Dyck sequences is shown. Also, we propose the method of branch ordering of a binary tree by means of only the corresponding Dyck sequence. We confirm that the method is in good consistency with the topological analysis of binary trees known as the Horton-Strahler analysis. Two types of central limit theorem are numerically confirmed, and the obtained results are expressed in simple forms. Furthermore, the proposed method is available for a wide range of the topological analysis of binary trees.

UR - http://www.scopus.com/inward/record.url?scp=70449567507&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449567507&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/42/41/415002

DO - 10.1088/1751-8113/42/41/415002

M3 - Article

AN - SCOPUS:70449567507

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 41

M1 - 415002

ER -