  The Steiner distance of vertices in a set $S$ is the minimum size of a
connected subgraph that contain these vertices. The sum of the Steiner
distances over all sets $S$ of cardinality $k$ is called the Steiner $k$-Wiener
index and studied as the natural generalization of the famous Wiener index in
chemical graph theory. In this paper we study the extremal structures, among
trees with a given segment sequence, that maximize or minimize the Steiner
$k$-Wiener index. The same extremal problems are also considered for trees with
a given number of segments.
