  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 theminimum 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.
