Admissible-cut Hopf algebra for subdivergence forests

Let $\mathcal{T}$ be the set of isomorphism classes of finite nonempty rooted trees. A rooted tree is a finite connected acyclic graph with one distinguished vertex, the root; each edge is oriented from the vertex nearer the root (the parent) to the vertex farther from the root (the child). A forest is a finite multiset of rooted trees, including the empty forest $1$. Let $H=\mathbb{Q}[\mathcal{T}]$ be the commutative polynomial algebra generated by $\mathcal{T}$; its monomial basis consists of forests, and the degree of a forest is its total number of vertices.

For a rooted tree $t$, an admissible cut is a subset $C$ of its edges meeting every root-to-leaf path in at most one edge. Removing the edges of $C$ leaves one component containing the root; denote it by $R_C(t)$. The components on the child side of the cut edges form a forest denoted by $P_C(t)$. For the empty edge cut, set $P_{\mathrm{\varnothing }}(t)=1$ and $R_{\mathrm{\varnothing }}(t)=t$. Define the coproduct on a tree generator by $\mathrm{\Delta }(t)=t\otimes 1+{\sum }_{C\text{ admissible}}{P}_C(t)\otimes R_C(t),$ where the sum includes the empty edge cut and retains multiplicities of distinct edge subsets. The term $t\otimes 1$ is the separate virtual full cut above the root. Extend $\mathrm{\Delta }$ multiplicatively to forests. Define $\epsilon (1)=1$ and $\epsilon (F)=0$ for every nonempty forest $F$. This specifies the rooted-tree subtraction model; it does not assert a canonical rooted-tree Hopf algebra or a graph-to-forest map for every Feynman graph.