Let $\mathcal{F}$ be the smallest family of (simple, finite) graphs such that:

- The empty graph is in $\mathcal{F}$.
- If $G \in \mathcal{F}$, then $G_{\bullet}$, the graph obtained from $G$ by adding another vertex $v$ connected by an edge to each vertex in $G$, is also in $\mathcal{F}$.
- If $G,H \in \mathcal{F}$, then $G \cup H \in \mathcal{F}$.

There are natural bijections between $\{G \in \mathcal{F}: \textrm{$G$ has $n$ vertices}\}$ and the set of rooted (but unlabeled) forests on $n$ vertices and between $\{G \in \mathcal{F}: \textrm{$G$ has $n$ vertices and is connected}\}$ and the set of rooted (but unlabeled) trees on $n$ vertices.

Does $\mathcal{F}$ have a name? I want to call it something like the set of "generalized threshold graphs" because the rules that define it are close to those of threshold graphs.