## Electronic Communications in Probability

### The Aldous-Shields model revisited with application to cellular ageing

#### Abstract

In Aldous and Shields (1988) a model for a rooted, growing random binary tree with edge lengths 1 was presented. For some $c>0$, an external vertex splits at rate $c^{-i}$ (and becomes internal) if its distance from the root (depth) is $i$. We reanalyse the tree profile for $c>1$, i.e. the numbers of external vertices in depth $i=1,2,...$. Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we say that nodes in depth $h+1$ are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senesced vertices for large $h$.

#### Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 43, 475-488.

Dates
Accepted: 19 October 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243986

Digital Object Identifier
doi:10.1214/ECP.v15-1581

Mathematical Reviews number (MathSciNet)
MR2733372

Zentralblatt MATH identifier
1226.60129

Rights