Our tree left last year about 2 days after the puppy came home. Since we cut a real one, and celebrate multiple December holidays here, and keep it up till epiphany (Jan 6th), we wait till the middle of the month to get one, usually.
Last year it was 2 days of baby gate on the living room. I expect it will be 3 weeks of no unsupervised dogs in the tree room... I have no idea what you do, other than enclosing the tree in an ex pen, if you can't shut the whole room. (I'd put an rx pen around the tree and gifts... they are like 60 bucks at walmart...)