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We prove that the chromatic symmetric function of any $n$-vertex treecontaining a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is,not a positive linear combination of elementary symmetric functions.Generalizing this, we also prove that the chromatic symmetric function of any$n$-vertex connected graph containing a cut vertex whose deletion disconnectsthe graph into $d\geq\log _2n +1$ connected components is not $e$-positive.Furthermore we prove that any $n$-vertex bipartite graph, including all trees,containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is notSchur-positive, namely not a positive linear combination of Schur functions. Incomplete generality, we prove that if an $n$-vertex connected graph has noperfect matching (if $n$ is even) or no almost perfect matching (if $n$ isodd), then it is not $e$-positive. We hence deduce that many graphs containingthe claw are not $e$-positive.

Schur and $e$-positivity of trees and cut vertices