Gromov-Hausdorff distance ($d_{GH}$)

For compact metric spaces $X$ and $Y$, the Gromov-Hausdorff distance iswhere the infimum runs over all metric spaces $Z$ and all isometric embeddings $\phi :X\to Z$ and $\psi :Y\to Z$. It measures how closely the two spaces can be placed inside one ambient metric space.

The Gromov-Hausdorff topology on isometry classes of compact metric spaces is the topology induced by the Gromov-Hausdorff distance.

A real tree is a metric space $(T,d)$ in which every pair $a,b\in T$ is joined by a unique arc and that arc is isometric to the real interval $[0,d(a,b)]$. Equivalently, it is a geodesic metric space with no nontrivial simple loops.

The multiplicity of $a$ in a real tree $T$ is the number of connected components of $T\setminus \{a\}$. A point of multiplicity one is a leaf, one of multiplicity two lies in the interior of an unbranched arc, and one of multiplicity at least three is a branch point.

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.