  We study Stanley's chromatic symmetric function (CSF) for trees when
expressed in the star-basis. We use the deletion-near-contraction algorithm
recently introduced in \cite{ADOZ} to compute coefficients that occur in the
CSF in the star-basis. In particular, one of our main results determines the
smallest partition in lexicographic order that occurs as an indexing partition
in the CSF, and we also give a formula for its coefficient. In addition to
describing properties of trees encoded in the coefficients of the star-basis,
we give two main applications of the leading coefficient result. The first is a
strengthening of the result in \cite{ADOZ} that says that proper trees of
diameter less than or equal to 5 can be reconstructed from their CSFs. In this
paper we show that this is true for all trees of diameter less than or equal 5.
In our second application, we show that the dimension of the subspace of
symmetric functions spanned by the CSF of $n$-vertex trees is $p(n)-n+1$, where
$p(n)$ is the number of partitions of $n$.
