The Aldous-Shields model revisited with application to cellular ageing
Katharina Best, University of Freiburg
Peter Pfaffelhuber, University of Freiburg
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 $h$.
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