Why the number of nodes with degree $d$ in a graph $G$ is equal to the number of copies of $d$-stars in $G$ that are not part of any $(d + 1)$ star in $G$ ?
Isn't it possible to have nodes with degree $d$ which don't belong to a $d$-star?
Let's think to a node with degree $d$ that seems to be the center of a star, but at least one couple of the fake star's vertices is connected and indeed it is not the center of a star....
For me a $d$ star is a subgraph of $d+1$ vertices and $d$ edges where there exists only edges from the center to the vertices, but not edges between vertices within a star. Other edges such the ones between vertices of different stars are allowed to exist.