On stability of weighted spanning tree degree enumerators

Paper [cherkashin2023stability] shows that the (vertex) spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial (id est is non-zero if all variables have positive imaginary parts) if and only if $G$ is distance-hereditary. In this note we generalize the result on weighted graphs.
This generalization allows us to define the class of weighted distance-hereditary graphs.