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 Krön 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.

我们证明每个图都有一个典型的树分解树，可以有效地区分所有主要缠结（包括末端和各种大型有限密集结构）。

在这里，“树的树分解”是一个比“树分解”稍弱的概念，但比“树状度量空间”表现得更好。这个定理是最好的，因为我们给出了一个例子，即在上述定理中，“树分解的树”不能被强化为“树分解”。

这意味着 Dunwoody 和 Krön 以及 Carmesin、Diestel、Hundertmark 和 Stein 的结果。除此之外，对于局部有限图，我们的结果给出了每个$克\in \mathbb{N}$有效区分所有k 可区分端的规范树分解。