It is shown that there exists a natural metric \(D\) on the set \[ T:= \{t\subseteq \mathbb{R}\mid \sup t< +\infty, \inf t\in t\} \] of bounded subsets of \(\mathbb{R}\) containing their infimum such that \((T, D)\) is a complete connected metric space which in addition satisfies the so-called four-point condition, that is, the particular condition required for this metric space being an \(\mathbb{R}\)-tree. It is this \(\mathbb{R}\)-tree we have dubbed the real tree. Due to its explicit description, a long list of assertions concerning its diverse properties can be established, including the following: (1) the set of (equivalence classes of) its ends corresponds in a canonical one-to-one way to the set \(E\) of subsets \(e \subset \mathbb{R}\) with \(\sup e<+ \infty\) enlarged by \(\varnothing\) (leading to a canonical valuated matroid structure of rank 2 defined on this set), and (2) for every \(t\in T\), the cardinality of the set of connected components of \(T\backslash \{t\}\) (i.e. the ``degree'' of \(t\)) coincides with the cardinality \(\# {\mathcal P}(\mathbb{R})\) of the set of all subsets of \(\mathbb{R}\).