Context : It has been established that Gamma-Ray Bursts ( GRBs ) are connected to Supernovae ( SNe ) explosions of Type Ib/c .

Aims : We intend to test whether the hypothesis of Type Ib/c SNe from different massive progenitors can reproduce the local GRB rate as well as the GRB rate as a function of redshift .
We aim to predict the GRB rate at very high redshift under different assumptions about galaxy formation and star formation histories in galaxies .

Methods : We assume different star formation histories in galaxies of different morphological type : ellipticals , spirals and irregulars , which have been already tested in self-consistent galaxy models reproducing both chemical and photometrical properties of galaxies .
We explore different hypotheses concerning the progenitors of Type Ib/c SNe : i ) single massive stars ( M > 25 M _ { \odot } , Wolf-Rayet stars ) , ii ) massive close binaries ( 12-20 M _ { \odot } ) and iii ) both Wolf-Rayet stars and massive binaries .
We conclude that the mixed scenario ( iii ) is preferable to reproduce the local Type Ib/c SN rates in galaxies and we adopt this scenario for comparison with the GRB rates .

Results : We find an excellent agreement between the observed GRB local rate and the predicted Type Ib/c SN rate in irregular galaxies , when a range for single Wolf-Rayet stars of 40-100 M _ { \odot } is adopted .
We also predict the cosmic Type Ib/c SN rate by taking into account all the galaxy types in an unitary volume of the Universe and we compare it with the observed cosmic GRB rate as a function of redshift .
By assuming the formation of spheroids at high redshift , we predict a cosmic Type Ib/c SN rate , which is always higher than the GRB rate , suggesting that only a small fraction ( 0.1-1 \% ) of Type Ib/c SNe become GRBs .
In particular , we find a ratio between the cosmic GRB rate and the cosmic Type Ib/c rate in the range 10 ^ { -2 } -10 ^ { -3 } , in agreement with previous estimates .
Finally , due to the high star formation in spheroids at high redshift , which is our preferred scenario for galaxy formation , we predict more GRBs at high redshift than in the hierarchical scenario for galaxy formation , a prediction which awaits to be proven by future observations .

Conclusions :