  We investigate, using purely combinatorial methods, structural and
algorithmic properties of linear equivalence classes of divisors on tropical
curves. In particular, an elementary proof of the Riemann-Roch theorem for
tropical curves, similar to the recent proof of the Riemann-Roch theorem for
graphs by Baker and Norine, is presented. In addition, a conjecture of Baker
asserting that the rank of a divisor D on a (non-metric) graph is equal to the
rank of D on the corresponding metric graph is confirmed, and an algorithm for
computing the rank of a divisor on a tropical curve is constructed.
