  When does a graph admit a tree-decomposition in which every bag has small
diameter? For finite graphs, this is a property of interest in algorithmic
graph theory, where it is called having bounded ``tree-length''. We will show
that this is equivalent to being ``boundedly quasi-isometric to a tree'', which
for infinite graphs is a much-studied property from metric geometry. One object
of this paper is to tie these two areas together. We will prove that there is a
tree-decomposition in which each bag has small diameter, if and only if there
is a map $\phi$ from $V(G)$ into the vertex set of a tree $T$, such that for
all $u,v\in V(G)$, the distances $d_G(u,v), d_T(\phi(u),\phi(v))$ differ by at
most a constant.
  A ``geodesic loaded cycle'' in $G$ is a pair $(C,F)$, where $C$ is a cycle of
$G$ and $F\subseteq E(C)$, such that for every pair $u,v$ of vertices of $C$,
one of the paths of $C$ between $u,v$ contains at most $d_G(u,v)$ $F$-edges,
where $d_G(u,v)$ is the distance between $u,v$ in $G$. We will show that a
graph $G$ admits a tree-decomposition in which every bag has small diameter, if
and only if $|F|$ is small for every geodesic loaded cycle $(C,F)$. Our proof
is an extension of an algorithm to approximate tree-length in finite graphs by
Dourisboure and Gavoille.
  In metric geometry, there is a similar theorem that characterizes when a
graph is quasi-isometric to a tree, ``Manning's bottleneck criterion''. The
goal of this paper is to tie all these concepts together, and add a few more
related ideas. For instance, we prove a conjecture of Rose McCarty, that $G$
admits a tree-decomposition in which every bag has small diameter, if and only
if for all vertices $u,v,w$ of $G$, some ball of small radius meets every path
joining two of $u,v,w$.
