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It’s not legal to intentionally damage a neighbor’s tree. Even if the roots of a tree encroach into your yard, if you damage or kill a tree that belongs to your neighbor, you can be held liable for the damage in most cases. Since herbicides can spread through the tree, causing damage to the whole tree and not just the roots that are on your side of the property line, you shouldn’t poison your neighbor’s tree. The other downside to this method is that if the tree dies, it can fall onto your property and cause more damage, so herbicides are best avoided in this circumstance.

No, definitely not. Pulling weeds by hand is the single most effective way to remove them because you're eliminating the entire plant—roots and all—so it won’t have a chance to grow or spread. Setting time aside a few times per week to pull weeds can keep them at bay. If the number of weeds gets overwhelming, then you can resort to other methods.

Weeds can grow through rubber mulch, but it depends on if you removed all the weeds in your garden beforehand. Weeds already in the soil can grow even when using rubber mulch. However, that doesn’t mean that your garden will be littered with rubber and weeds, but it does mean that the occasional weed might still pop up. Another way you can increase your weed suppression is by using landscape fabric as a second layer. 

Adding vinegar to your lawn increases the acidity of the area, which can kill quack grass present above the ground. However, putting vinegar in your yard will also kill the grass. Unfortunately, the below-ground quack grass rhizomes remain very much alive in the presence of vinegar and will soon sprout again.