A forest of rooted non-planar trees $t_{i}$ is a set (unordered collection) of rooted non-planar trees, i.e. of directed graphs with a distinguished vertex (root) such that there exists a unique path from the root to any other vertex.
Given a forest $F=t_{1}\cdots t_{n}$, its grafting is a rooted non-planar tree $B^{+}(F)$ obtained from $F$ by connecting all roots of the forest to a new vertex, which is declared the root of the resulting tree. For instance, given trees (A,B,(C,D)E)F (Newick format) and (G, H)I, their grafting is ((A, B, (C, D)E)F, (G, H)I)J. Ungrafting $B^{-}(t)$ is an operation on trees inverse to grafting, i.e. it turns a rooted tree into a forest by deleting the root.
Grossman-Larson product $*$ of forests $t_{1}\cdots t_{n}$ and $s_{1} \cdots s_{m}$ is defined here (page 33) in the following way:
If $t_{1}\cdots t_{m}$ is an empty forest, denoted as $\mathbf{1}$, then $$1 * s_{1}\cdots s_{m}=s_{1}\cdots s_{m}$$. If $t_{1}\cdots t_{n}$ is not empty, then $t_{1}\cdots t_{n} * s_{1}\cdots s_{m}$ is the sum of ungraftings of trees, obtained by grafting $s$ the forest $s_{1}\cdots s_{m}$ and then attaching roots of elements of the forest $t_{1}\cdots t_{n}$ to $s$ in all $n^{|s|}$ possible ways, where $s:=B^{+}(s_{1}\cdots s_{m})$.
An example given in the text is
The definition itself seems clear, but that "all $n^{|s|}$" part doesn't make sense to me. Suppose we have a tree $s=B^{+}(s_{1}\cdots s_{m})$ that contains $|s|$ vertices, and we have $n$ trees in the forest $t_{1}\cdots t_{n}$. Each of them we can attach to any of $|s|$ vertices independently, giving us $|s|^{n}$ choices. Is this a typo, or do I miss something?