For a simple graph G, the Gordon-Scantlebury index of G is equal to the number of paths of length two in G, and the Platt index is equal to the total sum of the degrees of all edges in G. In this paper, we study these indices in random plane-oriented recursive trees through a recurrence equation for the first Zagreb index. As n ∊ ∞, the asymptotic normality of these indices are given.