A tree decomposition of a graph $F$ is a tree $T$ with bags $\beta(t) \subseteq V(F)$ for $t \in V(T)$ such that every vertex of $F$ lies in some bag, both endpoints of every edge lie in a common bag, and, for every vertex $v$ of $F$, the nodes whose bags contain $v$ induce a connected subtree of $T$. Edge multiplicities play no role.

Let $\Delta \geq 1$. The $\Delta$-forestwidth $\mathrm{fw}_\Delta(F)$ of $F$ is the least $w \geq 0$ such that $F$ admits a tree decomposition $(T, \beta)$ with root $r$ such that $T$ has depth at most $\Delta$, $\beta(r) = \emptyset$, and $|\beta(t)| \leq w + 1$ for all $t \in V(T)$. Equivalently, deleting $r$ yields a forest decomposition of $F$ in which every root-leaf path has at most $\Delta$ nodes. This is a modification of the $\Delta$-treewidth, for which this forest decomposition is required to be a tree decomposition.

We formalise the statement $\mathrm{fw}_\Delta(F) \leq w$ as the existence of such a decomposition. A rooted tree of depth at most $\Delta$ is given by its finite set of nodes, its root $r$, and the parent map, extended by $\mathrm{parent}(r) = r$, such that $\mathrm{parent}^\Delta(t) = r$ for every node $t$. In such a tree, a set $U$ of nodes induces a connected subtree if, and only if, at most one node $t \in U$ is the root or has its parent outside $U$. As the root bag is empty, it suffices to require for every vertex $v$ that at most one node $t$ has $v \in \beta(t)$ and $v \notin \beta(\mathrm{parent}(t))$.