The biosphere is a closed cycle, we cannot probably use it to store carbon away by means of its own mechanism. Main problem is that we are adding new fossil carbon into it every day.
EDIT: as the sibling comment observes, burning the tree cannot be "awful", as all the carbon from that tree came from the atmosphere. Growing and burning a tree is the epitome of carbon-neutral thing to do. Now, burying a tree may have a positive (non-neutral) effect.
If it gets covered, and spends time in a cold, anaerobic environment, there will be nothing to return it to the atmosphere. Decomposition can eventually stall out entirely, allowing it (given enough time and compression) to form coal or oil.
I'm very interested in wood gas pyrolosis, a process that pulls burnable fuel from wood and produces almost pure carbon as a byproduct. The gas could be used to generate electricity or heat, and the carbon could be buried in old mines. I suspect the economics just aren't there for it yet, (or the process doesn't scale to the numbers we'd need for large scale de-carbonization).
I'm hopeful that with the advent of LED lit trees reliability goes up and there is less early waste of perfectly good plastic with a few busted lightbulbs.
Here in Australia I think Artificial Trees are the norm, I've never really thought about it before, Possibly because Christmas is in Middle of summer and real trees would be difficult to keep looking fresh.
I always felt a lot of cognitive dissonance around Christmas growing up all the songs and decorations are about snow and winter but the weather here is oppressively hot.
We have the same kind of disconnect around here (Brazil): all these people dressed as Santa Claus (heavy-looking winter clothing) in the middle of 40 degree Rio de Janeiro summer has always been an odd juxtaposition.
Article from last year: https://www.nola.com/gambit/article_5ab7222e-2e60-11ea-a845-....