First step, contact your towns building code and enforcement department. They will be able to tell you what is required, send out someone to take a look etc. I would not worry too much about whether or not the garage was permitted if it was built in 1951. The house has probably changed hands a few times which means that a plat or survey was generated and, even if the structure was over the property line or in an easement it probably met code or was grandfathered in.
I found this on line that might give you some leads as to where to begin and how to proceed: “What is self-help?
Property owners in every state have the right to trim the branches or roots of a neighbor’s tree that encroach onto their property, up to the property line, at their own expense. This right is called “self-help.” Self-help is an alterative to going to court. The rationale is that self-help prevents the wasteful use of the court system to resolve comparatively minor disputes. It’s a trade-off: you have the right to cut and remove the encroaching branches or roots of your neighbor’s tree, right away, at your own expense (i.e., use self-help), instead of having to hire a lawyer, start a lawsuit, and wait for the courts to sort it out. Using self-help saves you time and money, and keeps the courts from settling disputes between neighbors. In Minnesota, you have the option of using self-help OR going to court, when using self-help is not practical or reasonable. In most other states, self-help is the exclusive remedy.”
Evidently, and I would verify this locally, you can even cut the roots where they penetrate into your yard. This might cause damage and possibly even death of the tree but there ya’ go.
Good luck and keep us informed.