Fence issues are a common problem between neighbors, and issues may arise when a fence is on the boundary of both properties. Responsibility will depend on a few different factors.
Texas A&M explains that it often depends on who built the fence if it is on a boundary line.
In general, the obligation to maintain and repair a fence built on a boundary line between two properties is with whoever built the fence. If your neighbor built the fence, then you hold no responsibility for taking care of it. If you built the fence, then you assume all responsibility.
You should be aware, though, that if you refuse to assist with maintenance and car of the fence, then you have no ownership over it. However, if you do help your neighbor with maintaining it, then you do hold some ownership. You can agree to help with maintenance, but note that this creates a legally binding obligation. Any such agreement should go into writing and include all responsibilities and obligations.
Also, if the fence is not on a boundary line, then it belongs to the person whose property it is on. If you want to build a fence that will encroach on your neighbor’s property, then you must ask for permission. If you fail to do so, you could face trespassing charges.
Trees are often an issue around fence or property lines. The law states that the tree belongs to the person whose land on which the trunk grows. If it is your tree, you hold all responsibility. So, if your tree should grow over a fence and the branch breaks and damages the fence, then you would be responsible for the damages even if the fence is not yours. In addition, your neighbor has the right to trim the branches back to your property line.