We usually attach our tree to the wall, just to be on the safe side.
My parents have two cats and I have two more that I bring home with me. They're all adult cats and mine will be kept in a different room but you never know... One of our past cats decided to climb the tree once and it fell, of course.
We've had a hook on the wall ever since, we tie a string to the hook and the other end around the tree trunk. We also usually try to find a tree that's not too tall so we can place it on a stool, that way the cats won't do as much harm even if they play under the tree.
They love to crawl under the Christmas tree mat, lol. Otherwise they pretty much ignore the tree after the first day.