  This is a new and short proof of the main theorem of classical structure tree
theory. Namely, we show the existence of certain automorphism-invariant
tree-decompositions of graphs based on the principle of removing finitely many
edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main
ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We
extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem
on the structure of finitely generated groups with more than one end.
