You say "both share the cost of its upkeep or liability for its lack of upkeep, in proportion to how much of the trunk is on whose side of the line", but the 2006 edition of Nolo's _Neighbor Law_ says that in California the tree is %50 your neighbor's, even if only a little bit of the trunk is on his property. If the tree was originally only on your property, then the trunk grew to where it was also on your neighbor's property, your neighbor becomes co-owner.
So I'm a little confused here--proportional, or 50%?
Curious (and yes, I do have a tree problem),
Well, as in the law generally, there are two (or more) sides to every issue. The law is ambiguous, so the sides are fuzzy too, like a shaggy barked Eucalyptus. I hope it's not a Eucalyptus coming between you and your neighbor. There are no hard and fast rules. You are both responsible, true, and neither can remove or damage the tree without permission of the other. You may however, trim the branches on your side of the line, so long as you do not damage the structural integrity of the tree.
The reason these problems are not so commonly written about today is 1) most of these issues came up in the days when farmers planted windbreaks to protect the crops along their boundary lines and 2) many people today use the services of a trained mediator to resolve disputes.
That is what I would suggest if you are having a problem with a boundary tree. Your town or County may have a mediation service for just such disputes. If not, you and the neighbor need to agree to hire someone and split the costs.
I wish you good luck with your tree and if you need further advice, consult your local Bar Association for the name of a good tree lawyer in your area.