  Given a set $P$ of $n$ points in $\mathbb{R}^2$ and an input line $\gamma$ in
$\mathbb{R}^2$, we present an algorithm that runs in optimal $\Theta(n\log n)$
time and $\Theta(n)$ space to solve a restricted version of the $1$-Steiner
tree problem. Our algorithm returns a minimum-weight tree interconnecting $P$
using at most one Steiner point $s \in \gamma$, where edges are weighted by the
Euclidean distance between their endpoints. We then extend the result to $j$
input lines. Following this, we show how the algorithm of Brazil et al.
("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) that
solves the $k$-Steiner tree problem in $\mathbb{R}^2$ in $O(n^{2k})$ time can
be adapted to our setting. For $k>1$, restricting the (at most) $k$ Steiner
points to lie on an input line, the runtime becomes $O(n^{k})$. Next we show
how the results of Brazil et al. ("Generalised k-Steiner Tree Problems in
Normed Planes", arXiv:1111.1464) allow us to maintain the same time and space
bounds while extending to some non-Euclidean norms and different tree cost
functions. Lastly, we extend the result to $j$ input curves.
