# Mutations on a Random Binary Tree with Measured Boundary

## TL;DR

This paper models mutations on infinite random binary trees with boundary measures, characterizing the allelic partition and clonal structures, and introduces a coalescent point process representation for such trees.

## Contribution

It establishes a mapping of supercritical binary trees to coalescent point processes and characterizes the regenerative properties of the clonal boundary.

## Key findings

- The boundary measure converges to a uniform measure on the ultrametric tree.
- The clonal boundary forms a regenerative set with a specific structure.
- The clonal subtree dynamics form a Markovian increasing tree process.

## Abstract

Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles `co-existing at infinity'. We prove that any such tree can be mapped to a random, compact ultrametric tree called coalescent point process, endowed with a `uniform' measure on its boundary which is the limit as $t\to\infty$ of the properly rescaled counting measure of the population at time $t$. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.