  We introduce a two-parameter family of probability measures on spanning trees
of a planar map. One of the parameters controls the activity of the spanning
tree and the other is a measure of its bending energy. When the bending
parameter is 1, we recover the active spanning tree model, which is closely
related to the critical Fortuin--Kasteleyn model. A random planar map decorated
by a spanning tree sampled from our model can be encoded by means of a
generalized version of Sheffield's hamburger-cheeseburger bijection. Using this
encoding, we prove that for a range of parameter values (including the ones
corresponding to maps decorated by an active spanning tree), the
infinite-volume limit of spanning-tree-decorated planar maps sampled from our
model converges in the peanosphere sense, upon rescaling, to an
SLE$_\kappa$-decorated $\gamma$-Liouville quantum cone with $\kappa > 8$ and
$\gamma = 4/\sqrt\kappa \in (0,\sqrt 2)$.
