### Abstract

Original language | English |
---|---|

Article number | 87 |

Number of pages | 10 |

Journal | Electronic Communications in Probability |

Volume | 18 |

Early online date | 9 Nov 2013 |

DOIs | |

Publication status | Published - 2013 |

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### Cite this

*Electronic Communications in Probability*,

*18*, [87]. https://doi.org/10.1214/ECP.v18-2784

**Increasing paths in regular trees.** / Roberts, M; Zhuo Zhao, Lee.

Research output: Contribution to journal › Article

*Electronic Communications in Probability*, vol. 18, 87. https://doi.org/10.1214/ECP.v18-2784

}

TY - JOUR

T1 - Increasing paths in regular trees

AU - Roberts, M

AU - Zhuo Zhao, Lee

PY - 2013

Y1 - 2013

N2 - We consider a regular n-ary tree of height h, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of paths from the root to a leaf along vertices with increasing labels. We show that if α=n/h is fixed and α>1/e, the probability that there exists such a path converges to 1 as h→∞. This complements a previously known result that the probability converges to 0 if α≤1/e.

AB - We consider a regular n-ary tree of height h, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of paths from the root to a leaf along vertices with increasing labels. We show that if α=n/h is fixed and α>1/e, the probability that there exists such a path converges to 1 as h→∞. This complements a previously known result that the probability converges to 0 if α≤1/e.

UR - http://www.scopus.com/inward/record.url?scp=84887413686&partnerID=8YFLogxK

UR - http://ecp.ejpecp.org/article/view/2784/2364

UR - http://dx.doi.org/10.1214/ECP.v18-2784

U2 - 10.1214/ECP.v18-2784

DO - 10.1214/ECP.v18-2784

M3 - Article

VL - 18

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

M1 - 87

ER -