Electronic Communications in Probability

Increasing paths in regular trees

Matthew Roberts and Lee Zhao

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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 $\alpha = n/h$ is fixed and $\alpha > 1/e$, the probability that there exists such a path converges to $1$ as $h \to \infty$. This complements a previously known result that the probability converges to $0$ if $\alpha \leq 1/e$.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 87, 10 pp.

Accepted: 9 November 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60C05: Combinatorial probability 92D15: Problems related to evolution

evolutionary biology trees branching processes increasing paths

This work is licensed under a Creative Commons Attribution 3.0 License.


Roberts, Matthew; Zhao, Lee. Increasing paths in regular trees. Electron. Commun. Probab. 18 (2013), paper no. 87, 10 pp. doi:10.1214/ECP.v18-2784. https://projecteuclid.org/euclid.ecp/1465315626

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