oh, I've never heard of a birthday tree before...my daughter's birthday is Dec 22 and she came home on Christmas Eve Day. She does make a singular "birthmas" list every year though, rather than making two separate lists. 🙂
When they were born, their mother suggested to me we give them the option to celebrate the birthday in summer rather than the actual date...so that they wouldn't get combined birthday/Hanukkah gifts and miss out. Never did happen, though.
The Birthday Tree was a tradition we started as a bonding thing. For most of their life, though we are in frequent contact via phone and text and we see each other as often as we can, we were apart. This tradition was something we shared. I figured, since the world was putting up lights and celebrating, we might as well, too, because we had something to celebrate as well! So each year I find a new ornament that has something to do with the year just past: something we shared, or something they experienced, or some milestone they passed.
This really worked out particularly well when I moved in with my current wife. My daughter would come out for their birthday every other year, and my step-kids traditionally spend Christmas Eve here and Christmas Day at their dad's, so we have this combined celebration and no one feels like a fifth wheel.