  We investigate the late-time evolution of the Yang-Mills field in the
self-gravitating backgrounds: Schwarzschild and Reissner-Nordstr\"om
spacetimes. The late-time power-law tails develop in the three asymptotic
regions: the future timelike infinity, the future null infinity and the black
hole horizon. In these two backgrounds, however, the late-time evolution has
quantitative and qualitative differences. In the Schwarzschild black hole
background, the late-time tails of the Yang-Mills field are the same as those
of the neutral massless scalar field with multipole moment l=1. The late-time
evolution is dominated by the spacetime curvature. When the background is the
Reissner-Nordstr\"om black hole, the late-time tails have not only a smaller
power-law exponent, but also an oscillatory factor. The late-time evolution is
dominated by the self-interacting term of the Yang-Mills field. The cause
responsible for the differences is revealed.
