As far as I'm aware, apoptosis (the programmed death of cells) has nothing to do with the short tails seen in modern birds. Tail vertebrae do form in a developing chicken, but the number that appear is smaller than in dinosaurs such as T. rex - the number of segments in the tail has simply been reduced. Of the tail vertebrae that do appear, a few at the front are incorporated into a structure called the synsacrum, which consists of the pelvic bones and the adjacent vertebrae. (Because all the bones are fused together, the synsacrum looks a bit like a bony carapace in the pelvic region of the chicken skeleton.) The backmost few tail vertebrae also fuse together, to make a bladelike structure called the pygostyle. Apparently the pygostyle is colloquially known as the "parson's nose" or the "pope's nose", although I had never seen either term before I started looking at anatomy textbooks. In any case, there are a half-dozen or so individual, unfused tail vertebrae in between the synsacrum and the pygostyle, but they remain very short throughout growth. Behind the synsacrum, then, an adult chicken does have a free tail consisting of a few short vertebrae and then the fused pygostyle. The bony tail is obvious in a chicken skeleton, but in a living chicken it's obscured by flesh and feathers.