  Through direct thermodynamic calculations we have shown that different
classical entropies of two-dimensional extreme black holes appear due to two
different treatments, namely Hawking's treatment and Zaslavskii's treatment.
Geometrical and topological properties corresponding to these different
treatments are investigated. Quantum entropies of the scalar fields on the
backgrounds of these black holes concerning different treatments are also
exhibited. Different results of entropy and geometry lead us to argue that
there are two kinds of extreme black holes in the nature. Explanation of black
hole phase transition has also been given from the quantum point of view.
