What does appear to be true is that in 1902, when young Archie was just eight years old, he did sneak a tree into the White House and kept it upstairs in a closet. He supposedly had White House staff members help him find lights and ornaments, and he added a gift for each of his family members, including the Roosevelts' pets. On Christmas morning, he finally shared the surprise with his family, who were delighted by it. The president even let his children continue the tradition after that.
Here's what Roosevelt wrote in a letter to a friend about that Christmas:
Yesterday Archie got among his presents a small rifle from me and a pair of riding boots from his mother. He won’t be able to use the rifle until next summer, but he has gone off very happy in the riding boots for a ride on the calico pony Algonquin, the one you rode the other day. Yesterday morning at a quarter of seven all the children were up and dressed and began to hammer at the door of their mother’s and my room, in which their six stockings, all bulging out with queer angles and rotundities, were hanging from the fireplace. So their mother and I got up, shut the window, lit the fire (taking down the stockings of course), put on our wrappers and prepared to admit the children.