There is a paper in JSTOR, [The continuum random tree, I][1], by David Aldous, that is similar to your question.  That is the first of a three part series of papers.  However, it doesn't seem like Aldous' tree (or tree ensemble) can be the same as your tree.  Your tree would have infinite valence everywhere, because the harmonic series diverges.

  [1]: http://www.jstor.org/stable/2244250