  In a 1995 paper Richard Stanley defined $X_G$, the symmetric chromatic
polynomial of a Graph $G=(V,E)$. He then conjectured that $X_G$ distinguishes
trees; a conjecture which still remains open. $X_G$ can be represented as a
certain collection of integer partitions of $|V|$ induced by each $S\subseteq
E$, which is very approachable with the aid of a computer. Our research
involved writing a computer program for efficient verification of this
conjecture for trees up to 23 vertices. In this process, we also gather trees
with matching collections of integer partitions of a fixed number of parts. For
each $k=2, 3, 4, 5$, we provide the smallest pair of trees whose partitions of
$k$ parts agree. In 2013, Orellana and Scott give a proof of a weaker version
of Stanely's conjecture for trees with one centroid. We prove a similar result
for arbitrary trees, and provide examples to show that this result, combined
with that of Orellana and Scott, is optimal.
