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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 symmetricfunctions such that for each vertex coloring of $G$ there is exactly one ofthese summands. The question, whether $X_{G}(x_1,x_{2},\ldots)$ distinguishesnonisomorphic 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 Wagnerproved it for spiders and some caterpillars. We decompose a tree by separatingthe leafs and their neighbors and do the same to the remaining forest untilthere remains a forest with vertices of degree not greater than $1$. Fornonisomorphic trees $G$ and $H$ with the same number of vertices and specialproperties concerning their number of leafs and leaf neighbors for eachsubgraph in their leaf decomposition we prove that the chromatic symmetricfunction distinguishes the graphs. Our idea is to find independent partitionsof $G$ and $H$ with respectively a block of maximal cardinality for $G$ as wellas for $H$. These cardinalities are different for graphs $G$ and $H$ withspecial properties of their leaf decompositions. Additionally, we give explicitformulas for the cardinality of such a maximal block of an independentpartition of star connections and spiders.

On distinguishing special trees by their chromatic symmetric functions