TRIM THAT LIMB

Can you cut a neighbor’s tree if the branch is on your property?  Generally, YES, as long as long as you don’t kill the tree.
Furthermore, if a limb falls on your property because of the neighbor’s negligence, they have to deal with the damages and the fallen tree.  What constitutes “negligence” here?  If the tree is sick or diseased and the owner knows it and a limb falls on your house, that is negligence.  If the tree is infested with insects that damage the tree, the owner will likely be responsible for damage caused by fallen limbs because, again, that is negligence. 
But what if the tree is healthy.  In this case, the owner might be negligent, and thus responsible for damages cause by the tree, if the tree is not properly maintained or pruned.  Liability here would take a showing that the owner deliberately disregarded a situation, for example, where a tree on their property had a limb growing over a neighbor’s house.  The limb was allowed to grow too long and heavy compared to the rest of the tree.  As a result, it could be argued that the owner neglected the problem and was liable for damages caused by the limb when it fell.
If the tree is healthy though, and a limb falls due to a storm or wind, that will be the problem of the landowner where the limb fell.  The owner of the tree (where the trunk is located) is generally not liable for damages where a storm blows a limb or tree down.  Some municipalities call it “an act of God” meaning you can’t blame or hold anyone responsible.  
Are you sure who owns the tree?  This one seems simple at first.  Ownership is based on where the trunk of the tree is.  On your property?, your tree.  On your neighbor’s property?, their tree.   But what about trees right on the boundary.  Here both owners share ownership and responsibility for the tree.  In this situation, the parties should attempt to come to an agreement regarding the fate of a tree.  You really should have a meeting of the minds before a tree on a boundary line is cut down.  In general though, each owner is basically responsible for the part of the tree on their side of the property line.
Can you cut down a tree on your own whenever you want?  Not everywhere.  Some locals, usually upscale neighborhoods or small touristy towns it seems, have requirements that you get a permit to cut down a tree.  Some call the requirement a “Tree Preservation Order”.  Be very aware that you need to check on this before cutting down a tree, even in your own yard.
For the most part and to answer the most commonly asked question about trees, you do not have to let a neighbor’s overhanging branch endanger your home.  As long as you can do so without killing the tree, you can usually cut it in order to protect your property.
Got it?  I wood think so.