‎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.

‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b≥ 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

The first variable Zagreb index of graph G is defined as Ml1 (G)=SxÎv (G) d (v)2l, where l is a real number and d (v) is the degree of vertex v. In this paper, some lower and upper bounds for the expected value and distribution function of this index in random increasing trees (recursive trees, plane-oriented recursive trees and binary increasing trees) are given.

Our main objective is to study some qualitative behavior of the solutions of the difference equation xn+1 = gxn−k + (axn + bxn−k) / (cxn − dxn−k), n = 0, 1, 2, ..., where the initial conditions x−k,..., x−1, x0 are arbitrary positive real numbers and the coefficients, a, b, c and d are positive constants, while k is a positive integer number.