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  "body": "\n## Statement\n\nLet *G* be a group acting on a tree *X* without inversions. Let **A** be the quotient graph of groups and let *v* be a base-vertex in *A*. Then *G* is isomorphic to the group π\u003csub\u003e1\u003c/sub\u003e(**A**, *v*) and there is an equivariant isomorphism between the tree *X* and the Bass–Serre covering tree $\\tilde {\\mathbf A}$. More precisely, there is a group isomorphism σ: *G* → π\u003csub\u003e1\u003c/sub\u003e(**A**, *v*) and a graph isomorphism $j:X\\to \\tilde {\\mathbf A}$ such that for every *g* in *G*, for every vertex *x* of *X* and for every edge *e* of *X* we have *j*(*gx*) = *g* *j*(*x*) and *j*(*ge*) = *g* *j*(*e*).\n\nThis result is also known as the *structure theorem*.\n\nOne of the immediate consequences is the classic Kurosh subgroup theorem describing the algebraic structure of subgroups of free products.\n",
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