  We compute bounds in the expected Ollivier curvature for the Brownian
continuum random tree $\mathcal{T}_{\mathbb{e}}$. The results indicate that
when the scale dependence of the Ollivier curvature is properly taken into
account, the Ollivier-Ricci curvature of $\mathcal{T}_{\mathbb{e}}$ is bounded
above by every element of $\mathbb{R}$ for almost all points of
$\mathcal{T}_{\mathbb{e}}$. This parallels the well-known result that every
continuum tree is a $CAT(K)$ space for all $K\in\mathbb{R}$.
