Asymptotic normality of the Zagreb index of random $b$-ary recursive trees

The $b$-ary recursive trees model is one of simple families of increasing trees. In this work, the Zagreb index $Z_n$ of a random $b$-ary recursive tree of size $n$ is studied. As $n\to\infty$, the asymptotic normality of $Z_n$ is established through the martingale central limit theorem, as well as the asymptotic expressions of the mean and variance of $Z_n$ are given.