Reduced words in a graph of groups

Fix a graph of groups $\mathcal{G}$ and a maximal subtree $T$. A graph-of-groups word is an expression

$g_0{e}_1{g}_1\cdots e_n{g}_n,$

where $(e_1,\dots ,e_n)$ is an edge path in the underlying graph and each $g_j$ lies in the vertex group at the intermediate vertex.

Such a word is closed when its edge path starts and ends at the same vertex. A word of edge length $0$ is closed at the vertex containing its unique coefficient.

Such a word is reduced when no cancellation pattern $e_{j+1}={\bar{e}}_j$ occurs with the intervening coefficient $g_j$ lying in the edge-group image ${\alpha }_{{\bar{e}}_j}(G_{e_j})$. Tree edges remain in this word notation to record movement between vertex groups, even though their edge symbols represent the identity in ${\pi }_1(\mathcal{G},T)$.