I need help in characterizing the classes of graphs that results from taking the complementary of a tree, i.e., the graph that results from removing the edges of a tree from a complete graph. More formally, let $T=(V,E)$ be an n-vertex tree with vertex set V and edge set E. Are there known results on the classes of graphs defined by $\overline{T}$?

There are two trivial cases. If $T={S}_{n}$, i.e., is a star tree (one single vertex has degree $n-1$ and the other vertices have degree 1), we have that $\overline{{S}_{n}}=\{v\}\cup {K}_{n-1}$, i.e., $\overline{{S}_{n}}$ is a graph with an isolate vertex (degree 0) and a $(n-1)$ vertex clique. I've got the feeling that $\overline{{P}_{n}}$ (where ${P}_{n}$ is an n-vertex path graph) has a precise characterization but I can't put a name to it.

If there are no (or few) results about general T, can we say something about $\overline{T}$ if T belongs to a class of trees, for example caterpillar trees (trees in which the removal of all leaves produces a path graph), lobster trees (trees in which the removal of all leaves produces a caterpillar tree), ...?

Any help will be appreciated. Thank you.