We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

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Let {X , X i , i ≥ 1} be i.i.d. random variables, S k be the partial sum andIn this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that