You can cut anything that hangs over into your property, so long as it does not result in the death of the tree or plant. You are responsible for all of the costs.
Sending a certified letter may place the homeowner on notice, but it doesn't automatically make the neighbor responsible for cleanup if the tree later falls. The neighbor will be responsible for cleanup if he or she is found to be negligent in some way. Negligence in these types of cases is typically found when a property owner knew, or reasonably should have known, that the tree or plant was likely to cause damage, and did nothing to mitigate it.
In this case, if the tree is diseased or otherwise obviously unhealthy, a certified letter can be used later to help prove the requisite "notice" if you later have to sue to get a fallen tree cleaned up. However, if the tree is not diseased or unhealthy, and is just leaning, well, unless you have some specialized knowledge (you are an arborist or something), this may not be sufficient to create negligence. A leaning, but otherwise healthy tree, that falls does not necessarily mean that the neighbor was negligent. It might, but it will depend on all of the facts of the situation.
In other threads, where people have been advised to notify their neighbors in order to create notice for a later tort, neighbor's roots or vines have been causing actual damage, or have the potential to cause damage in a place where the plant owner would not know of the damage (roots in a cellar, for example). In those types of cases, it is imperative to put the neighbor on notice by sending a letter.
In the case of a tree, the neighbor is "on notice" if the tree is diseased or damaged such that a reasonable person would notice the disease or damage -- whether or not a letter is sent. The letter can help prove the notice, but it is not required.
The flip side, like I stated above, is that a letter cannot create the negligence if none exists. I can't send a letter to my neighbor saying that an otherwise healthy tree (or what appears to be healthy to a reasonable person) might cause damage if it fell, and then be able to sue for negligence later if the tree does fall, say, in a storm. In other words, a certified letter alone cannot turn an "act of god," for which you would be responsible for cleaning up and repairing, into negligence, which would shift the burden to your neighbor.
Again, if you have specialized knowledge or training, or were to hire someone with specialized knowledge or training, and then determine that the tree is problematic, then a letter may be able to create negligence later on. But a lay person cannot do this.