Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability $1-q$ and the opposite value with probability $q$, where $q \in [0,1]$. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of $q$ for which the optimal reconstruction method has a probability of error bounded away from $1/2$. We also show that the probability of error is bounded by a constant times $q$. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.