## Electronic Communications in Probability

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

### Increasing paths in regular trees

#### Abstract

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

**Source**

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

**Dates**

Accepted: 9 November 2013

First available in Project Euclid: 7 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1465315626

**Digital Object Identifier**

doi:10.1214/ECP.v18-2784

**Mathematical Reviews number (MathSciNet)**

MR3141796

**Zentralblatt MATH identifier**

1306.60128

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 60C05: Combinatorial probability 92D15: Problems related to evolution

**Keywords**

evolutionary biology trees branching processes increasing paths

**Rights**

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

#### Citation

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