It's my understanding that the birds evolved from the early bird-hipped dinosaurs, of which T. Rex is a later member. The first known bird, Archaeopteryx, is from the late Jurassic, which predated the appearance of T.Rex by tens of millions of years.

We have to be careful of making the conceptual mistake, and I catch myself doing it all the time, that evolution is a line flowing into the future from the past. It's really a tree, or a dendritic drainage pattern, with many branches, parallel lines, and dead ends.

The sky may already have been filled with birds just before the Cretaceous event, and for all we know, T.Rex may have had feathers.