Number of maximal rooted trees in preferential attachment model via stochastic approximation

We study the asymptotic behavior of the number of maximal trees in the preferential attachment model. In our model, we consider a sequence of graphs built by the following recursive rule. We start with the complete graph on $m+1$ vertices, $m>1$. Then on the $n+1$ step, we add vertex $n+1$ and draw $m$ edges from it to different vertices from $1,\ldots,n$, chosen with probabilities proportional to their degrees plus some positive parameter $\beta$. We prove the convergence speed for the number of maximal trees in such a model using the stochastic approximation technique.