  The Brownian map is a model of random geometry on the sphere and as such an
important object in probability theory and physics. It has been linked to
Liouville Quantum Gravity and much research has been devoted to it. One open
question asks for a canonical embedding of the Brownian map into the sphere or
other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has
been shown to be "equivalent" to the Brownian map but the exact nature of the
correspondence (i.e.\ embedding) is still unknown. In this article we show that
any embedding of the Brownian map or continuum random tree into $\mathbb{R}^d$,
$\mathbb{S}^d$, $\mathbb{T}^d$, or more generally any doubling metric space,
cannot be quasisymmetric. We achieve this with the aid of dimension theory by
identifying a metric structure that is invariant under quasisymmetric mappings
(such as isometries) and which implies infinite Assouad dimension.
  We show, using elementary methods, that this structure is almost surely
present in the Brownian continuum random tree and the Brownian map. We further
show that snowflaking the metric is not sufficient to find an embedding and
discuss continuum trees as a tool to studying "fractal functions".
