  Stanley asked whether a tree is determined up to isomorphism by its chromatic
symmetric function. We approach Stanley's problem by studying the relationship
between the chromatic symmetric function and other invariants. First, we prove
Crew's conjecture that the chromatic symmetric function of a tree determines
its generalized degree sequence, which enumerates vertex subsets by cardinality
and the numbers of internal and external edges. Second, we prove that the
restriction of the generalized degree sequence to subtrees contains exactly the
same information as the subtree polynomial, which enumerates subtrees by
cardinality and number of leaves. Third, we construct arbitrarily large
families of trees sharing the same subtree polynomial, proving and generalizing
a conjecture of Eisenstat and Gordon.
