The inverse problem for the Steiner-Wiener index of trees

For a connected graph $G$ and a set $S\subset V(G)$, the Steiner distance $d_G(S)$ is the minimum number of edges in a connected subgraph of $G$ containing $S$.

The Steiner-Wiener $k$ index is defined by $\mathrm{SW}_k(G) = \sum_{S\subset V(G), |S|=k} d_G(S)$.

We study the inverse problem for this invariant restricted to trees: for fixed $k$, which positive integers occur as $\mathrm{SW}_k(T)$ for a finite tree $T$?

We prove that all sufficiently large positive integers occur as $\mathrm{SW}_k(T)$ for some finite tree $T$ if and only if $k$ is even.

For odd $k$, we further show that the set of attainable values has asymptotic density of order $k^{-\delta}(\log k)^{-3/2}$, where $\delta$ is the Erdős-Tenenbaum-Ford constant.