Extremal values for Steiner distances and the Steiner $k$-Wiener index

Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend several theorems on the middle parts and extremal values of trees from their regular distance variants to their Steiner distance variants. More specifically, we show that for a tree $T$, the Steiner $k$-distance, Steiner $k$-leaf-distance, and Steiner $k$-internal-distance are all concave along a path. We also calculate distances between the Steiner $k$-median, Steiner $k$-internal-median, and Steiner $k$-leaf-median. Letting the Steiner $k$-distance of a vertex $v\in V(T)$ be ${\text{dd}}_k^T(v)$, we find bounds based on the order of $T$ for the ratios $\frac{{\text{dd}}_k^T(u)}{{\text{dd}}_k^T(v)}$, $\frac{{\text{dd}}_k^T(w)}{{\text{dd}}_k^T(z)}$, and $\frac{{\text{dd}}_k^T(u)}{{\text{dd}}_k^T(y)}$ where $u$ and $v$ are leaves, $w$ and $z$ are internal vertices, and $y$ is a Steiner $k$-centroid. Also, denoting the Steiner $k$-Wiener index as ${\mathsf{S}\mathsf{W}}_k(T)$, we find upper and lower bounds for $\frac{{\mathsf{S}\mathsf{W}}_k(T)}{{\text{dd}}_k^G(v)}$. The extremal graphs that produce these bounds are also presented.