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Yes, suggesting companion planting and mulching to minimize new growth.

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.

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. 

Purchase a selective herbicide that targets clover and similar weeds but doesn't take out your grass. You can also keep it from growing with corn meal gluten or remove clover from your lawn manually at the start of the season as part of your lawn maintenance checklist.