It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in <https://en.wikipedia.org/wiki/Real_tree> admits an isometric embedding into the Banach space $\ell\_1(\Gamma)$ for a set $\Gamma$ of sufficiently large cardinality (the space $\ell\_1(\Gamma)$ is defined as the space of real-valued functions on $\Gamma$ with countable support and norm $||f||=\sum\_{\gamma\in\hbox{supp} f}|f(\gamma)|$). My question is:


Is this known? If "Yes", I would appreciate references.


P.S. I like the proof of this fact suggested by Yves de Cornulier (see below). I also agree with Bill Johnson that the fact looks like "well known" and most probably has been published somewhere. If someone knows a reference, I would be very thankful for it (I cannot accept the second answer, but I would be happy to upvote it.)



Embedding of real trees into $\ell_1(\Gamma)$
It's true: there's an isometric embedding of every real tree $T$ on some $\ell^1$-space, i.e., a set $A$ and a map $f:T\to \ell^1(A)$ that is an isometric embedding, that is, satisfies $\|f(x)-f(x')\|=d(x,x')$ for all $x\in T$.


I don't know a reference; here's a proof. Let $(x\_t)\_{t<\alpha}$ be an enumeration of points on the real tree $T$ by an ordinal $\alpha$, assume that the convex hull of them is $T$. 


Let $T\_\beta$ be the closed convex hull of $\{x\_t:t<\beta\}$ (so $T\_\alpha=T$). Let us construct by induction an isometric embedding $f\_\beta:T\_\beta\to\ell^1(\beta)\subset\ell^1(\alpha)$, so that whenever $\beta\le\gamma$ then $f\_\gamma$ extends $f\_\beta$.


For $\beta=1$, $T\_\beta=\{x\_0\}$, and we just prescribe $f\_\beta(x\_0)=0$. For $\beta=\gamma+1>1$ a successor ordinal, $T\_\beta$ is the convex hull of $T\_\gamma\cup\{x\_\gamma\}$. Let $y\_\gamma$ be the projection of $x\_\gamma$ on $T\_\gamma$. So $$T\_\beta=T\_\gamma\cup [y\_\gamma,x\_\gamma].$$ We define $f\_\beta$ so as to extend $f\_\gamma$ on $T\_\gamma$, and prescribe $f\_\beta(y)=f\_\gamma(y\_\gamma)+d(y,y\_\gamma)e\_\beta$ for every $y\in [y\_\gamma,x\_\gamma]$, where $(e\_t)$ is the canonical basis of $\ell^1(\alpha)$ and $d$ is the distance in the tree. This is well-defined (note that $f\_\gamma(y\_\gamma)$ is defined twice) and defines an isometric embedding. Finally, for $\beta$ a limit ordinal, then $T\_\beta$ is the closure of $T'\_\beta=\bigcup\_{\gamma<\beta}T\_\gamma$, we just define $f\_\beta$ on $T'\_\beta$ by saying its graph is the union of graphs of all $f\_\gamma$ for $\gamma<\beta$; this is an isometric embedding into a complete metric space, so extends uniquely to an isometric embedding of the closure $T\_\beta$ into $\ell^1(\beta)$.


So $f\_\alpha$ is the required isometric embedding.


