When our oldest children were old enough to read gift tags on packages they began to sort and stack packages under the Christmas tree into piles according to the name on the tag. That was annoying to me as they compared piles of presents, who had the biggest package, guessing what was in the package, etc. It was distracting to the focus we wanted to have in our home and seemed to bring out the "gimmies'.
To solve the problem we began randomly assigning each member of the family a different reindeer name each year and they would not find out which reindeer they were until Christmas. The piling, comparing and guessing stopped completely! It is a system we have used successfully for about 25 years on the gifts that are from Mom and Dad.
Another idea for handling presents under the tree that I really like is the tradition of each child receiving 3 gifts, representing the gifts from the wisemen. One gift could be something useful, one could be something fun that is wanted and one a book or something related to a skill or hobby that the child has interest in.