I seek a proof of the following result due to Tits:
Theorem: A path-connected $0$-hyperbolic metric space is a real tree.
Do you know any proof or reference?
Might this be what you are looking for?
J. Tits, A "theorem of Lie-Kolchin" for trees, Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, pp. 377–388.
I finally found the theorem in a document by Steven N. Evans: Probability and Real Trees (theorem 3.40).
Moreover, path-connected can be replaced by connected.