Every regular infinite tree set is represented by the edge tree set of a tree-like space, and the regular tame ones are exactly the edge tree sets of ordinary trees.
Tangles and the Mona Lisa
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abstract
We show how an image can, in principle, be described by the tangles of the graph of its pixels. The tangle-tree theorem provides a nested set of separations that efficiently distinguish all the distinguishable tangles in a graph. This translates to a small data set from which the image can be reconstructed. The tangle duality theorem says that a graph either has a certain-order tangle or a tree-structure witnessing that this cannot exist. This tells us the maximum resolution at which the image contains meaningful information.
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Representations of infinite tree-sets
Every regular infinite tree set is represented by the edge tree set of a tree-like space, and the regular tame ones are exactly the edge tree sets of ordinary trees.