On the probability of occurence of labelled subtrees of a randomly labelled tree
For a positive integer b, consider an infinite binary (rooted) tree in which every node has one parent (excepting the root) and exactly b children. Let every node v be assigned with a random variable \(X_ v\), such that \(\{X_ v\}_{v\in V}\) are independent identically distributed binary variables: \[ P(X_ v=0)=1-p,\quad P(X_ v=1)=p. \] Denote by \(\pi_ n(p)\) the probability that each node of level less than n has at least one child assigned with 0 and at least one child assigned with 1. As clearly \(\pi_{n+1}(p)\leq \leq \pi_ n(p)\) there must exist a value \[ \pi_*(p)=\lim_{n\to \infty}\pi_ n(p). \] The main result is the following assertion. Theorem. If \(b\leq 4\) then \(\pi_*(p)=0\) for all \(p\in <0,1>;\) if \(b\geq 5\) then there exists \(p_ c\in (0,\frac12)\) such that \(\pi_*(p)=0\) if \(p<p_ c\) or \(p>1-p_ c,\) and \(\pi_*(p)>0\) for \(p\in <p_ c,1-p_ c>.\)
scientific article; zbMATH DE number 4214029