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.