Chromatic Symmetric Functions and Polynomial Invariants of Trees

Stanley asked whether a tree is determined up to isomorphism by its chromaticsymmetric function. We approach Stanley's problem by studying the relationshipbetween the chromatic symmetric function and other invariants. First, we proveCrew's conjecture that the chromatic symmetric function of a tree determinesits generalized degree sequence, which enumerates vertex subsets by cardinalityand the numbers of internal and external edges. Second, we prove that therestriction of the generalized degree sequence to subtrees contains exactly thesame information as the subtree polynomial, which enumerates subtrees bycardinality and number of leaves. Third, we construct arbitrarily largefamilies of trees sharing the same subtree polynomial, proving and generalizinga conjecture of Eisenstat and Gordon.