  By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under
CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it
has kappa many branches for some kappa strictly between omega_1 and
2^{omega_1}. In this paper we prove that, assuming the existence of one
inaccessible cardinal,
  (1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist
Kurepa trees and there are no Jech--Kunen trees,
  (2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees
with omega_3 many branches exist.
