Probabilities of events related to common predecessors of two vertices in a generalized model of recursive trees

We say that a random tree $T_n$ with $n$ vertices and $n-1$ edges is a generalized recursive one if either $n=1$, or $n>1$ and $T_n$ is the result of linking some $n$th vertex to some vertex of a random recursive tree $T_{n-1}$. The probability to choose a particular vertex is defined by some sequence $\{\alpha_i\colon \alpha_i>0\}_{i=1}^\infty$. We study the probabilities of some events related to common predecessors of vertices.