  We generalise structure tree theory, which is based on removing finitely many
edges, to removing finitely many vertices. This gives a significant
generalization of Tutte's tree decomposition of 2-connected graphs into
3-connected blocks. For a finite graph there is a structure tree that contains
information about $k$-connectivity for any $k$. The theory can also be applied
to infinite graphs that have more than one vertex end, i.e. ends that can be
separated by removing a finite number of vertices. This gives a generalization
of Stallings' structure theorem for groups with more than one end.
