Explain why many barbarian tribes converted to christianity during the fourth century.

The reason why so many tribes converted during the fourth century was because many of them were under the influence of the Roman Empire, that established Christianism as its official religion during the reign of Emperor  Constantine the Great which lasted from 306 to 337.

Constantine first decide to stop the persecution of Christians in the territories of the Empire. Then, in 313, he enacted the Edict of Milan that stated that Christian worship was not a crime anymore. He also converted to the new religion, that replaced the previous politeistic deities.