  We study the structure of trees minimizing their number of stable sets for
given order $n$ and stability number $\alpha$. Our main result is that the
edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars,
each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor
\frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two
distinct stars, and the centers of these stars form a stable set of the tree.
