This is not a straightforward yes/no answer. If you put a Christmas tree on a large compost heap, then over time, yes of course it will decompose, like everything else that was once alive. However it will most likely take a very long time to break down, considering that the base of the trunk alone could be six inches wide. Do you really want to wait several years for your compost?
So, assuming you were planning to use your compost this year, then putting a whole Christmas tree straight into the compost is perhaps not the way to go. So what other options are available?
Well you could strip the branches off and maybe just use these in the compost, although even then it will take a very long time to break down.
Or you could use a wood chipper or mechanical shredder to shred / chip the tree. You could definitely use these chips in your compost. They still might take up to a year to decompose though. Some people might have concerns about the straw pine needles making your compost too acidic, but this is unlikely unless you have very little else in your compost.
Alternatively, these wood chips could be used straightaway in the garden as an excellent mulch.
Hopefully this has given you a few ideas about what to do with your Christmas tree once the festive season has passed.