A note on distinguishing trees with the chromatic symmetric function

For a tree T , consider its smallest subtree T ◦ containing all vertices of degree at least 3. Then the remaining edges of T lie on disjoint paths each with one endpoint on T ◦. We show that the chromatic symmetric function of T determines the size of T ◦, and the multiset of the lengths of these incident paths. In particular, this generalizes a proof of Martin, Morin, and Wagner that the chromatic symmetric function distinguishes spiders.

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