The key here (in my thinking) is that the poster has already admitted knowledge that HIS tree has "has damaged limbs which are hanging over his (neighbors) shed."
Here's an excerpt from my first reference:
"Traditionally, property owners were not responsible for damage caused by falling tree limbs and other natural occurrences on their property. However, they were responsible for damage caused by artificial conditions, such as a loose board from your lumber pile being carried by the wind through your neighbor's plate glass window. The current trend suggests that the courts are applying an ordinary standard of care to measure negligence in both cases. This means that maintaining your property in good condition is an important protection against a negligence suit.
For example, if your trees have visible rot, you should cut them down or trim rotted limbs before they can fall on your neighbor's property. Trees should be maintained well enough that, short of a tornado or hurricane, the wind won't blow things from your place over to your neighbor's."
And from the 3rd:
"If your neighbor's tree is unsound and threatens your property, your neighbor is liable to you for any damage that occurs. The test is whether the tree owner knew or should have known that damage was likely. A tree owner is not expected to be a tree expert, but she is expected to recognize obvious symptoms of a problem, such as the unseasonal lack of leaves, a dead limb, visible decay, or a tree leaning dangerously to one side. Courts consider these trees to be a nuisance. Urban foresters sometimes consider these trees to be a hazard. A "hazard tree" is a tree with a defect plus a target, like the swingset in your yard, your garage, or your house. If the potential for damage is foreseeable and if the tree owner fails to take corrective action, the courts will likely hold the owner legally responsible for damage caused to others' property.
What should you do if your neighbor fails to address a problem tree?
· Give your neighbor written notice about the danger you foresee and ask her to fix the problem.
· Bolster your claim with an expert opinion about the tree's condition.
· Take photographs of the threatening condition.
· Mediate, using a neutral third party, to resolve the impasse.
· As a last resort, you can sue your neighbor. Small claims court is intended for claims up to $7,500, and lawyers aren't needed.
· Don't take the law into your own hands. Don't engage in stealth pruning or tree removal while your neighbor is gone. You risk being sued or arrested for trespassing or ordered to pay your neighbor money or "damages" to compensate her for the loss."
Normally, if this were a healthy tree, I would agree that the tree owner would probably not be liable. However, in this case, the writer has already admitted that the subject limbs are 'damaged'.