  We revisit the \textsc{$k$-Secluded Tree} problem. Given a vertex-weighted
undirected graph $G$, its objective is to find a maximum-weight induced subtree
$T$ whose open neighborhood has size at most $k$. We present a fixed-parameter
tractable algorithm that solves the problem in time $2^{\mathcal{O}(k \log
k)}\cdot n^{\mathcal{O}(1)}$, improving on a double-exponential running time
from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from
a single vertex, our algorithm grows a $k$-secluded tree by branching on
vertices in the open neighborhood of the current tree $T$. To bound the
branching depth, we prove a structural result that can be used to identify a
vertex that belongs to the neighborhood of any $k$-secluded supertree $T'
\supseteq T$ once the open neighborhood of $T$ becomes sufficiently large. We
extend the algorithm to enumerate compact descriptions of all maximum-weight
$k$-secluded trees, which allows us to count the number of maximum-weight
$k$-secluded trees containing a specified vertex in the same running time.
