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    Trees with Given Stability Number and Minimum Number of Stable Sets

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    We study the structure of trees minimizing their number of stable sets for given order nn and stability number α\alpha. Our main result is that the edges of a non-trivial extremal tree can be partitioned into n−αn-\alpha stars, each of size ⌈n−1n−α⌉\lceil \frac{n-1}{n-\alpha} \rceil or ⌊n−1n−α⌋\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.Comment: v2: Referees' comments incorporate
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