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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.

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

For nonempty compact subsets $A,B$ of a metric space $(Z,d)$, the Hausdorff distance isFor compact metric spaces $X,Y$, the Gromov-Hausdorff distance iswhere $\phi$ and $\psi$ range over isometric embeddings into a common metric space $Z$.

The collection of compact real trees is not compact in the Gromov-Hausdorff topology. Indeed, the intervals $T_n=[0,n]$ are compact real trees andTheir diameters are unbounded, so $(T_n)$ has no convergent subsequence in the Gromov-Hausdorff topology.

Choose a root $o\in T$ and finite sets $F_n\subset T$ whose union is dense, arranging that $F_n$ is a $2^{-n}$-net and $F_n\subset F_{n+1}$. Let $T_n$ be the finite subtree spanned by $o$ and $F_n$. A depth-first contour traversal of $T_n$, recording distance from $o$, gives a continuous excursion $g_n$ whose real tree encoded by an excursion is $T_n$.

The traversals may be chosen compatibly: when passing from $T_n$ to $T_{n+1}$, insert the new branch traversals into small time intervals at their attachment points. Since every new component has height at most $2^{-n+1}$, choose the time changes so thatAfter harmlessly taking a faster sequence of nets, these errors are summable. Hence $(g_n)$ is uniformly Cauchy and converges uniformly to a continuous $g:[0,1]\to {\mathbb{R}}_{+}$ with $g(0)=g(1)=0$.

The net property gives $d_{GH}(T_n,T)\to 0$. By the stated continuity of excursion coding, $d_{GH}(T_{g_n},T_g)\to 0$. Since $T_{g_n}$ is isometric to $T_n$, uniqueness of limits in the Gromov-Hausdorff distance implies that $T_g$ is isometric to $T$. This proves the excursion coding theorem for compact real trees.