A real tree is a metric space in which every pair is joined by a unique arc and that arc is isometric to the real interval . Equivalently, it is a geodesic metric space with no nontrivial simple loops.
The multiplicity of in a real tree is the number of connected components of . 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 be continuous with , and setThe function is a pseudometric. Its metric quotientis the real tree encoded by .
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.
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