The green Christmas tree was put up in the formal livingroom, shaped like a bowling alley. Anything wrapped went under that tree as soon as it was wrapped. My parents always went wayyyyyy overboard for Christmas, and as the family grew, so did the stack of gifts "under" the tree. To the point where they would continue in something like six feet all the way around the tree. People are often overwhelmed at the sight, until they're reminded that this loot is for three generations.
Gift baskets and such that contained food or perfumed substances didn't belong there, but unless it was obvious an item contained food or something else the dogs and/or cat would find irresistible we'd learn the hard way.
I knew someone with a horror story about a huge gift box of various cheeses that the giver had chosen to wrap and hand over well before Christmas without any warning to the recipient. Who had a Lab puppy. That wound up hospitalized from excess cheese consumption in one night. To this day, Patty teases that dog with, "Want some cheeeeeeeeeeeeeeeeeeeeeese?"
Our Santa gifts were a whole 'nother story, and magically appeared in the family room with the tacky silver tinsel tree (yes, I want one now, but I think I want it in purple), sorted by kid.
In my home, wrapped gifts will go under the tree as soon as it goes up, and as soon as they're wrapped. The stuff we buy each other as gifts that were meant for Christmas but needed sooner (like massage units for a desk chair), will have their empty boxes put under the tree as reminders that they really were part of the gift list.