  It is known that for every $\alpha \geq 1$ there is a planar triangulation in
which every ball of radius $r$ has size $\Theta(r^\alpha)$. We prove that for
$\alpha <2$ every such triangulation is quasi-isometric to a tree. The result
extends to Riemannian 2-manifolds of finite genus, and to
large-scale-simply-connected graphs. We also prove that every planar
triangulation of asymptotic dimension 1 is quasi-isometric to a tree.
