The Aldous-Shields model revisited (with application to cellular ageing)

In Aldous and Shields (1988), a model for a rooted, growing random binary tree 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. For c>1, we reanalyse the tree profile, 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 assume that nodes in depth h+1 are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senescent vertices for large h.

💡 Research Summary

The paper revisits the Aldous‑Shields random binary tree model, focusing on the regime where the branching rate parameter (c) exceeds one. In the original construction, each external (leaf) vertex at depth (i) splits into two new external vertices at depth (i+1) at an exponential rate (c^{-i}). When a split occurs the vertex becomes internal. The authors first derive explicit closed‑form expressions for the expected number of external vertices at each depth, denoted (X_i(t)). By applying martingale techniques and Laplace transforms they obtain
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