On Urysohn's $\mathbb{R}$-tree

In the short note of 1927, Urysohn constructed the metric space $R$ that is nowhere locally separable. There is no publication with indications that $R$ is a (noncomplete) $\mathbb{R}$-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed $\mathbb{R}$-trees isometric to $R$ unaware of the paper by Urysohn. In this paper the author considers various constructions of the $\mathbb{R}$-tree $R$ and of the minimal complete $\mathbb{R}$-tree of valency c including $R$, as well as the characterizations of $\mathbb{R}$-trees, their properties, and connections with ultrametric spaces.