Beginning with our first child's first Christmas, we did not place gifts under the tree prior to Christmas eve ( and then, we put them there after the kids were asleep).
Instead, we put books around the tree. All the books were Christmas-themed, or winter-themed, though not all were religious. They included Golden books, wordless picture books, lovely stories, humorous books, serious books, poetry books, long books, books with just a few pages, The Night Before Christmas (beautifully illustrated), The Polar Express and my favorite, A Small Miracle by Peter Collington.
At first they were those baby-proof, chew-proof safe soft books. As the kids grew older we added more age-appropriate books. Usually we bought a book or two a year, but each year we put every book out, even the baby books. So now our collection extends from "Baby's Christmas Bathtub Fun Book" and Max Lucado's "The Christmas Candle". I bring the books out the night that we set up the tree., and I put them away in a box when we take the tree down and store them away so they're special.
My kids are in their 20s now. They still look at the books, and when they stop by with friends, their friends love sitting by the tree and laughing over the childish stories and sometimes sitting quietly and reading the books. So under our tree, until Christmas eve, are just books. It's my favorite tradition.