Real tree encoded by an excursion

Let $g:[0,1]\to {\mathbb{R}}_{+}$ be continuous with $g(0)=g(1)=0$, and setThe function $d_g$ is a pseudometric. Its metric quotientis the real tree encoded by $g$.

Every compact real tree is isometric to a real tree encoded by an excursion. One construction takes finite subtrees spanning successively finer finite nets, performs depth-first contour traversals of those subtrees, and chooses compatible time parameterizations. The contour functions have a uniformly convergent subsequence, and continuity of excursion coding in the Gromov-Hausdorff distance identifies the limiting coded tree with the original tree.