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Yes, the same rules discussed above apply to your neighbor: They can trim one of your trees as long as they’re only cutting the portion that extends into their property. Your neighbor is not allowed to access your property without permission, nor can they cut past the property line. 

It is possible to cut down a tree on your property yourself, but it is generally not recommended unless specific conditions are met. You should only consider this project if the tree is small (e.g., under 4-5 inches in diameter and short enough that you don't need a ladder to reach the branches), and you are experienced in handling a chainsaw. The tree must also be located far from any power lines, fences, homes, or other structures. Even for small trees, the work can be dangerous, and it's always advisable to have a spotter to ensure the tree falls correctly and no branches fall on you.

For any tree that is large, requires a ladder to access, or is near electrical lines or structures, you should always hire a professional. Attempting to remove a large tree yourself is extremely dangerous, poses a risk of serious injury, and could result in significant damage to your property.

Before any tree removal, it's also important to check your local and state regulations to ensure the tree is not historic or otherwise protected by your city.

For safety reasons, leaning trees are best removed by a professional, as they may often tear away from the hinge, causing you to lose control over where the tree falls. If you choose to cut down a tree leaning slightly, avoid cutting off root protrusions before you start. Make your hinge a little longer than usual by cutting it deeper (up to 30% of the diameter) and applying a wedge into the felling cut on the side the tree is leaning. Stand away from the direction the tree leans while making your felling cut.

A tree can grow back from its stump, but full regrowth is unlikely (especially for more stubborn conifer trees). Instead, trees are more likely to send up a number of shoots from the dead stump, which will try to develop into primary stems over time. In other words, it will start to look increasingly bushy. The goal of killing and drying out a tree stump is to prevent that from happening, as well as make the stump easier to completely remove.