‎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.

We study the limiting distribution of the degree of a given node in a scaled attachment Random recursive tree, a generalized Random recursive tree, which is introduced by Devroye et. al (2011). In a scaled attachment Random recursive tree, every node i is attached to the node labeled [iXi] where X0,…, Xn is a sequence of i.i.d. Random variables, with support in [0, 1) and distribution function F. By imposing a condition on F, we show that the degree of a given node is asymptotically normal.

‎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.

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.

Introduction: Reconstruction of the oral cavity with a flap design containing the buccal mucosa and buccinator muscle but excluding the facial artery and vein is the topic of these case reports.Case Reports: This article uses Random pattern vertically oriented partial thickness buccinator myomucosal flap for intraoral reconstruction in two cases. The first was for lining the mandibular anterior vestibule in a trauma patient. The second was for oral side coverage of bone graft in special cleft patient. In both patients, this flap survived and good bone coverage with non-keratinized mucosa was obtained.Conclusion: Thin long buccal myomucosal flap not including facial artery and vein can survive.