The proof is that they can tell you the number of leaves (again) after you have secretly removed some. Since only you know the number you removed, if the differ

The proof is that they can tell you the number of leaves (again) after you have secretly removed some. Since only you know the number you removed, if the difference between their counts matches your number, it is likely that they can indeed count the leaves on the tree. It is possible they have correctly guessed, which is why you repeat the challenge until you are convinced.