I admire the practical desire to not duplicate toys, items and experiences with your kids. They probably don't need two of the exact same toy or need to go to the zoo with mom one day then back the next day with dad. But some things will be duplicated out of necessity (your kids have clothes, beds, food, etc at both houses right?).
Other things will be similar but not the same. For example, if your ex-wife takes the kids to the park to play, does that mean you are henceforth prohibited from taking the kids to the park? I should hope not. You can both take them to the park and bond with your kids. And I'd bet that how you do the park is a bit different from how your ex does it.
So if celebrating Christmas is important to you, by all means, set up your own tree. You shouldn't restrict how you live life with your children just because your ex did it first. Live your life according to what's important to you. Let your kids bond with you over those things. Some of those things are probably important to both you and your ex. And that's ok. That just means your kids might get two Christmases with two loving parents that they will remember fondly for years to come. And more positive experiences like that will be very good for them.
Personally, I can't imagine celebrating Christmas without some kind of tree. For our family, we always got ornaments to commemorate different things we did, like family vacations, births of children, etc. And decorating the tree with those ornaments has always been a very important part of Christmas. So if I were in your shoes, a second Christmas tree would be non-negotiable. It would be time to relive memories with the kids and make new ones.