A tree in our neighbor's yard hangs into our yard, over the patio. We’d like to trim the tree so that we can build our succah on our patio according to halacha. We know from previous experience that our neighbors will want us to use a company that is more expensive than our preferred company.

Q: What if they don’t allow us to trim it as much as we need to? Do we have to use our neighbor's company if they won’t pay the difference in price between their choice and our choice?

A: Bava Basra (27b) teaches that if a tree extends into a neighbor’s field, the neighbor may trim the branches up until the height needed for a person to raise the ox-goad he uses to drive his plowing ox.  If the tree’s branches are very dense (i.e. a carob or sycamore tree), he may cut off all the branches that extend over the property line.  If the property is an irrigated field and harmed by any shade, he may cut down all branches that extend over the property line.  These rulings are also recorded in Shulchan Aruch (Choshen Mishpat 155:26). The principle that emerges from these halachos is that one has the right to protect his property. If a neighbor’s tree causes damage or inhibits use of his property, one has the right to remove the branches.  Therefore, if the neighbor’s tree prevents you from erecting your sukkah in the preferred location, you may cut down the protruding branches.

Regarding the payment: that is your responsibility. Even though it’s your neighbor’s property that’s damaging yours, he is not obligated to pay for damage inflicted by his property that he does not cause. Since a tree grows of its own accord, it’s not the tree owner’s responsibility to pay for the removal of branches. In the event that the tree owner would prefer a more expensive company than you would choose, the difference between those two prices is his responsibility.

You must bear in mind that you may only trim those branches that prevent you from building a kosher sukkah. You may not instruct the tree trimmers to trim the trees to the property line if that is not necessary for you to be able to build your sukkah.