Broadcasting induced colourings of random recursive trees and preferential attachment trees

TL;DR

This paper analyzes two-colourings of various random trees, including preferential attachment and recursive trees, using urn models and combinatorics to describe their limiting distributions and structural properties.

Contribution

It introduces a comprehensive probabilistic framework for colourings in random trees, extending previous bond percolation results with new distributional insights.

Findings

Limiting distributions for colour counts and subtree sizes

Distributional results for leaves and fringe subtrees

Analysis of the largest monochromatic root subtree

Abstract

In this work we consider random two-colourings of random linear preferential attachment trees, which includes random recursive trees, random plane-oriented recursive trees, random binary search trees, and a class of random $d$-ary trees. The random colouring is defined by assigning the root of the tree the colour red or blue with equal probability, and all other vertices are assigned the colour of their parent with probability $p$ and the other colour otherwise. These colourings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of P\'olya urns, we prove limiting distributions, after proper rescalings, for the number of vertices of each colour, the number of monochromatic subtrees of each colour, as well as the number of leaves and fringe subtrees with two-colourings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root. The description of the limiting distributions extends previous work on bond percolation in random preferential attachment trees.