  One of the major outstanding conjectures in the study of chromatic symmetric
functions (CSF's) states that trees are uniquely determined by their CSF's.
Though verified on graphs of order up to twenty-nine, this result has been
proved only for certain subclasses of trees. Using the definition of the CSF
that emerges via the Frobenius character map applied to $\mathbb{C}[S_n]$, we
offer new algebraic proofs of several results about the CSF's of trees.
Additionally, we prove that a "parent function" of the CSF defined in the group
ring of $S_n$ can uniquely determine trees, providing further support for
Stanley's conjecture.
