  We define a minimization problem for paths on planar graphs that, on the
honeycomb lattice, is equivalent to the exploration path of the critical site
percolation and than has the same scaling limit of SLE_6. We numerically study
this model (testing several SLE properties on other lattices and with different
boundary conditions) and state it in terms of spanning trees. This statement of
the problem allows the definition of a random growth process for trees on two
dimensional graphs such that SLE is recovered as a special choice of boundary
conditions.
