TITLE: Characterizations of metric trees
QUESTION [8 upvotes]: Let $X$ be a geodesic space. Then the following conditions are equivalent:

For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$.
No subset of $X$ is homeomorphic to $S^1$.
$X$ is simply connected and its topological dimension (small inductive) equals $1$.
Every geodesic triangle is isometric to a tripod.
$X$ is $0$-hyperbolic in the sense of Gromov.
Intersection of any two closed balls is a closed ball or an empty set.
For every Lipschitz maps $\gamma:S^1\to X$ and $\pi:X\to\mathbb{R}^2$,
$$
\int_{S^1}(\pi\circ\gamma)^*(x\, dy)=0.
$$

A metric space that satisfies any of the above equivalent conditions is known as the metric tree or an $\mathbb{R}$-tree.
The problem is that it is very difficult to find a single place where one could find proofs of such equivalences. Thus my question is:

Question. Is there a single paper that where one would find seld-contained proofs of such characterizations of metric trees? I do not necessarily mean exactly the same characterizations. Just many equivalent characterizations.

I wanted to ask my student to write a survey paper that would cover in particular all such proofs. I think it would be a useful reference for people working in analysis on metric spaces, and for him it would be a good way to learn this material, but if there is already a good reference for such characterizations, then perhaps it would be a pointless task. This is why I am asking that question.

REPLY [2 votes]: Some equivalent characterizations of metric trees can be extracted from Theorem 8 of the paper "Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces" by Dress (1984) DOI link. The focus of the theorem is more on subsets of metric trees and injective hulls/tight spans, but the proof is self-contained.