My neighbor has a very big tree with a very big canopy overhanging about a quarter of my backyard. As far as I am concerned, the tree blocks out a good amount of sunlight to my yard, and its roots are invading what little soil I have, making it difficult for me to plant vegetables or what not. As far as the city is concerned, I am within my legal rights to trim off all branches that overhang my property--a big job for me, to be sure, and doable only in early winter once the tree becomes bare; and such solution in itself does not address the invading roots issue. As far as my neighbor is concerned, this would put the tree off-balance. My neighbor is interested in removing the tree altogether (roots and all) and replacing it with a smaller type of tree--one that does not grow so large but would still provide some shade to her backyard. I actually would prefer my neighbor's idea, but I'm afraid that her solution is logistically and financially unrealistic for her and ultimately would only serve to delay implementation of any solution. What would be the best solution for me/for me and my neighbor?

Where is Solomon when you need him?
Obviously the value and use of the tree in question is low from your point of view.
Although there is cost involved in tree removal (which you can ascertain by calling for bids), it is unclear what value might be lost.
At my own home, vegetable gardening is impractical due to shade, but the value of large native (Ponderosa Pine) trees is greater than the value of vegetables. My solution is to grow a garden elsewhere (because I can). Sometimes we don't realize the value of something until it is gone or replaced by something less. I don't mean to sound preachy, but sometimes it's worth considering other options. Of course the condition of the tree, its usefulness and its species will also affect its value.
If you & your neighbor opt to remove the tree, you might wish to have the same company do both the tree removal and the root removal for both properties. Somehow you will have to agree on sharing the cost.
If your neighbor finds removal is unaffordable, can you both live with the other "solution" which you presented (cut back limbs, roots?)?
Best of luck and neighborly relations.

The first step would seem to be to get several estimates for the work. Be sure to get separate prices for the tree removal and the root problems. Bear in mind that there are several options for stump/root removal. See the Technical Report on stump removal on this web site.
Stump Removal

Once you have these estimates, talk with your neighbor to see how you can arrange for the job. Maybe between the two of you, you can manage the cost of the job. Bear in mind here that though it is your neighbor’s tree, you have an interest in its removal. You should be prepared to help with the cost, as it will be to your benefit.

You are fortunate that your neighbor is willing to talk about it and perhaps remove the tree. Too many of these situations become contentious and lead to hard feelings.

--
Russ Carlson, RCA

Thank you for your suggestions. After additional discussion with my neighbor, trimming appears to be the most reasonable option. Now, I can't imagine that the roots of a tree would *retreat* to line up with the edge of a trimmed canopy, but I do imagine that the roots would cease from radiating further for as long as the canopy is kept trimmed. Am I correct in supposing this?

To the degree that branches are trimmed, there will be a reduction in energy supplied to the roots by the top of the plant. However, it is not possible to accurately predict which roots will be more or less affected, nor how far they will extend as a result. Roots extend in a search for water and minerals in a soil environment which has sufficient exchange of gases (O2, CO2). Physical barriers may hinder root growth direction. Select and careful root pruning might be considered, if the roots are intolerable, but it will affect the rest of the tree.