  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.

Probability;Combinatorics