  We find explicit formulas for the probabilities of general boundary visit
events for planar loop-erased random walks, as well as connectivity events for
branches in the uniform spanning tree. We show that both probabilities, when
suitably renormalized, converge in the scaling limit to conformally covariant
functions which satisfy partial differential equations of second and third
order, as predicted by conformal field theory. The scaling limit connectivity
probabilities also provide formulas for the pure partition functions of
multiple $\mathrm{SLE}_\kappa$ at $\kappa=2$.
