A note on spaces of asymptotic dimension one
  Let $X$ be a geodesic metric space with $H_1(X)$ uniformly generated. If $X$ has asymptotic dimension one then $X$ is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus $g \ge 2$ and one boundary component is at least two. 
Koji Fujiwara, Kevin Whyte