Puzzle n°40 (*) - A very special "best in show"

Last week-end, a very untypical cat show was trying a new way of selecting the "best in show" (the most beautiful cats of the show) : a bit like a tennis tournament... First, feline judges compared all cats two at a time. Each time, they chose the one they found best conforming to its breed standard. Then, once this completed for all cats, they went on comparing all the winners of the first round, two by two, choosing the best one among the two each time. And they went on and on... every round, only half the cats remained in the challenge. This lasted until they reached the much-awaited final...

How many comparisons did they do, knowing that there were 128 cats present at the show ?

... I'm giving up ...