Canonical trees of tree-decompositions

We prove that every graph has a canonical tree of tree-decompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently. Here trees of tree-decompositions' are a slightly weaker notion than tree-decompositions’ but much more well-behaved than tree-like metric spaces'. This theorem is best possible in the sense that we give an example that trees of tree-decompositions’ cannot be strengthened to `tree-decompositions’ in the above theorem. This implies results of Dunwoody and Kr"on as well as of Carmesin, Diestel, Hundertmark and Stein. Beyond that for locally finite graphs our result gives for each $k\in \mathbb{N}$ canonical tree-decompositions that distinguish all $k$-distinguishable ends efficiently.