  In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set
of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the
goal is to compute a Steiner tree that spans all the terminals and contains at
most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum
edge-length of the Steiner tree is minimized, where the length of a tree edge
is the Euclidean distance between its two endpoints. The problem is
well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)}
n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies
that the problem is fixed-parameter tractable (FPT). This settles an open
question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that
the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach
can be generalized to the problem with $\ell_p$ metric for any rational $1 \le
p \le \infty$, or even other metrics on $\mathbb{R}^2$.
