... every year for the last 5 years, our Christmas tree has fallen over at least once ... and ours just did.

WIth LO going to be here soon, I predict that this is the last year we have a live-dead tree. Artificial trees are a little more balanced and probably don't fall over.
Re: 2009: The last year for a live-dead CHristmas tree ...
Look into a bigger tree stand. The big plastic stands work really well. I have never had a tree fall over in 6 years.
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Ditto this. Our fake tree always fell over because the cats would try to climb it, but no problems with a live one. . .but to each their own.
We've done the big plastic ones, the cast iron ones, the stabilizer stands, and they've all fallen over. I think the problem is that DH isn't patient enough to stabilize the tree enough when he puts it in the stand.