The eccentric sequence of a connected graph \(G\) is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of \(G\) is the sum of the distances between all unordered pairs of vertices of \(G\). The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index \(W^{\lambda }\) for \(\lambda >0\) and \(\lambda <0\), and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter.

We also present similar results for the \(k\)-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set \(A\subseteq V(G)\) is the minimum number of edges in a subtree of \(G\) whose vertex set contains \(A\), and the \(k\)-Steiner Wiener index is the sum of distances of all \(k\)-element subsets of \(V(G)\). As a corollary, we obtain a sharp lower bound on the \(k\)-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.

连通图\（G \）的偏心序列是其顶点的偏心率的非递减序列。的Wiener指数\（G \）是所有无序对顶点之间的距离的总和\（G \） 。本作者最近确定了在具有给定偏心序列的所有树木中最小化维纳指数的独特树木。在本文中，我们证明这些结果不仅适用于维纳指数，而且适用于一类称为维纳型指数的基于距离的拓扑指数。这个类的特定情况包括超Wiener指数，该指数Harary，广义Wiener指数\（W ^ {\拉姆达} \）为\（\拉姆达> 0 \）和\（\拉姆达<0 \），以及互为补充的维纳指数。我们的结果暗示并统一了给定阶数和直径的树木在这些Wiener型指数上的已知界限。

对于具有给定偏心距序列的树木的\（k \）- Steiner Wiener指数，我们也给出了相似的结果。集合\（A \ subseteq V（G）\）的Steiner距离是\（G \）子树的顶点的最小边数，该子树的顶点集包含\（A \）和\（k \） - Steiner Wiener指数是\（V（G）\）的所有\（k \）个元素子集的距离之和。作为推论，我们获得了具有给定顺序和直径的树木的\（k \）- Steiner Wiener指数的尖锐下界，并确定在哪种情况下极值树是唯一的，从而纠正了文献中的错误。