‎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.

If G is a connected graph with vertex set V, then the eccentric connectivity index of G, xc (G), is defined as ånÎV (G) deg (n) ecc (n) where deg (n) is the degree of a vertex n and ecc (n) is its eccentricity. In this paper we show some convergence in probability and an asymptotic normality based on this index in two classes of random Bucket recursive trees.

Bucket recursive trees are an interesting and natural generalization of ordinary recursive trees and have a connection to mathematical chemistry. In this paper, we give the lower and upper bounds for the moment generating function and moments of the multiplicative Zagreb indices in a randomly chosen Bucket recursive tree of size n with maximal Bucket size b≥ 1. Also, we consider the ratio of the multiplicative Zagreb indices for different values of n and b. All our results reduce to the ordinary recursive trees for b =1.

This paper reviews Polya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • Polya-Eggenberger’s urn• Bernard Friedman’s urn• Generalized Polya urns• Extended urn schemes• Invertible urn schemesConnections to random trees are surveyed. Numerous applications to trees common in computer science are discussed, including:• Binary search trees• Fringe-balanced trees• m-ary search trees• 2–3 trees• Paged binary trees• Bucket quad trees• Bucket k–d treesThe applications also include various types of recursive trees:• Standard recursive trees• Pyramids• Plane-oriented recursive trees• Phylogenetic trees• Bucket recursive trees• SproutsLimit distributions, and phase changes therein are presented within the unifying theme of Polya urn models.

The main goal of this paper is to study the modified $F$-indices (modified first Zagreb index and modified forgotten topological index) of random $m$-oriented recursive trees (RMORTs). First, through two recurrence equations, we compute the mean and the variance of these indices in our random tree model. Second, we show four convergence in probability based on these indices. Third, the asymptotic normalities, through the martingale central limit theorem, 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.

In this paper we study the Zagreb index in Bucket recursive trees containing Buckets with variable capacities. This model was introduced by Kazemi in 2012. We obtain the mean and variance of the Zagreb index and introduce a martingale based on this quantity.

In order to investigate the effect of different parameters on dimensions of scour hole Downstream of a flip Bucket, extensive experiments were conducted. Effect of parameters like: radius of Bucket, drop height, tail water depth and unit discharge have been investigated. It was observed that dimensions of scour hole is a function of radius of Bucket, drop height, tail water depth and unit discharge. Dimensionless equations for prediction of scour hole dimensions are introduced.

Flip Bucket are one of the most important types of Energy depreciations and it play the role of energy depreciation in Spillway. In this research, the hydraulic flow characteristics of a flip Bucket are investigated with FLUENT model. In this study, three types of computational grid (coarse, intermediate, and fine) were used to discriminate the simulation environment. In order to simulate the flow, the k-ε (Standard, RNG, Realizable) and k-ω (standard and SST) models were used. Also, in order to find the best wall function, two types, standard wall and non-equilibrium wall function were investigated. The comparison criterion in this study is also the trajectory profile of jet water. The results show that the fine computational grid, the input speed condition for the flow input boundary, and the output pressure for the boundaries that are in contact with the air, provide the best possible results. Also, the standard wall function is chosen for the effect of the wall function, and the turbulent model k-ε (Standard) has the most consistent results with experimental results. When the jet is getting closer to end of basin, the computational results increase with the numerical results of their differences. The lower profile of the water jet has less sensitivity to the hydraulic jet profile than the hydraulic jet profile. In the pressure test, it was also found that the results show that the numerical values of the pressure in the lower landing number differ greatly in experimental results.