In general, if the property holds for all planar graphs of "f" faces, any change to the graph that creates an additional face while keeping the graph planar would keep "v" − "e" + "f" an invariant. Since the property holds for all graphs with "f" = 2, by mathematical induction it holds for all cases. Euler's formula can also be proved as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both "e" and "f" by one, leaving "v" − "e" + "f" constant. Repeat until the remaining graph is a tree; trees have "v" =  "e" + 1 and "f" = 1, yielding "v" − "e" + "f" = 2, i. e., the Euler characteristic is 2.