It seems like the complete binary tree with edge weights decaying exponentially (say halving) at each level should have a bounded doubling constant (lambda = 3?).

The union of k paths is difficult to analyze unless you have some bound on the diameter of the tree, or at least a minimum length for each path, hence the gamma-bad k-comb definition.

The result Brian presented is for metrics which (1) have a small doubling
constant and (2) are trees. The intersection of these two groups seems
very small indeed! If any of you can come up with an example of such a
metric that is substantially different from a straight line, I'd be very
interested to see it. (eg. Consider a tree which is a union of k paths;
how does the doubling constant depend on k, roughly?)

Here is a proof for a lemma about the doubling constant of k-combs that was omitted from Gupta, Krauthgamer and Lee's paper "Bounded Geometries, Fractals, and Low-distortion Embeddings" and wouldn't fit in my presentation.

http://morrison.ucsd.edu/~bmcfee/notes-bounded.pdf

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