  The mating of trees approach to Schramm-Loewner evolution (SLE) in the random
geometry of Liouville quantum gravity (LQG) has been recently developed by
Duplantier-Miller-Sheffield (2014). In this paper we consider the mating of
trees approach to SLE in Euclidean geometry. Let $\eta$ be a whole-plane
space-filling SLE with parameter $\kappa>4$, parameterized by Lebesgue measure.
The main observable in the mating of trees approach is the contour function, a
two-dimensional continuous process describing the evolution of the Minkowski
content of the left and right frontier of $\eta$. We prove regularity
properties of the contour function and show that (as in the LQG case) it
encodes all the information about the curve $\eta$. We also prove that the
uniform spanning tree on $\mathbb Z^2$ converges to $\mathrm{SLE}_8$ in the
natural topology associated with the mating of trees approach.
