  In 1995, Stanley introduced the well-known chromatic symmetric function
$X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric
functions such that for each vertex coloring of $G$ there is exactly one of
these summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishes
nonisomorphic trees with the same number of vertices, is still open in general.
For special trees it has already been shown. In 2008, Martin, Morin and Wagner
proved it for spiders and some caterpillars. We decompose a tree by separating
the leafs and their neighbors and do the same to the remaining forest until
there remains a forest with vertices of degree not greater than $1$. For
nonisomorphic trees $G$ and $H$ with the same number of vertices and special
properties concerning their number of leafs and leaf neighbors for each
subgraph in their leaf decomposition we prove that the chromatic symmetric
function distinguishes the graphs. Our idea is to find independent partitions
of $G$ and $H$ with respectively a block of maximal cardinality for $G$ as well
as for $H$. These cardinalities are different for graphs $G$ and $H$ with
special properties of their leaf decompositions. Additionally, we give explicit
formulas for the cardinality of such a maximal block of an independent
partition of star connections and spiders.
