The Wikipedia article about the Planar Separator Theorem states that it is possible to use a hierarchy of separators to construct a tree decomposition for a planar graph and moreover provides an $\mathcal{O}(\sqrt{n})$ upper bound for the tree width of an $n$ vertex planar graph.I have difficulties in understanding the hierarchy and moreover in checking that the resulting binary tree is a tree decomposition. I would really appreciate if you could help me or point to a reference that elaborates on it.
Planar separator theorem and tree decomposition
graph theory;planar graphs