This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikvári (2010) [4] and Csikvári (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and ℓ, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length ℓ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having the maximum coefficients in absolute value among the trees of order n which are neither a path nor a star.

该贡献对树的Wiener指数，封闭步道数，某些图多项式（邻接多项式，Laplacian多项式，边覆盖多项式和独立多项式）的系数进行了广泛的研究。Csikvári（2010）[4]引入了广义树移位，它保留了树的顶点数量。应用广义树移位和递归关系，我们扩展了Csikvári（2010）[4]和Csikvári（2013）[5]的工作。使用统一的方法，我们得到以下主要结果：首先，对于所有n和ℓ，我们描述具有最大（分别为最小）维纳指数的唯一树和具有最大（分别为最小）闭合数的唯一树。长走ℓ在n阶的树木中，它们既不是路径也不是星星。其次，我们描述了唯一树，其邻接多项式（分别为Laplacian多项式，独立多项式）在n阶树中分别具有绝对值的最大（最小）系数，它们既不是路径也不是星形。最后，我们确定了所有n个顶点覆盖树的边覆盖多项式的绝对值最小的树，并且还确定了在n个既不是路径也不是星形的树中绝对值最大的唯一树。