Tree roots are the most common causes of sewer line backups or blockages. They will either break the pipe altogether or invade the inside of the pipe through cracks or joints to initially hinder flow and then to ultimately block it altogether.
It's usually almost impossible to know when a sewer line will cause a problem before it happens without an actual camera inspection. Tree root problems are very slow and then seem to cause a problem unexpectedly when the roots reach a level sufficient enough to be noticed.
To open up pipes again one of the most common ways is with a snake. It will be sent down the sewer line chewing up the roots as it goes along. This is a temporary solution however if the pipe is damaged enough because the roots will grow back in over time. Sometimes roots in the pipe are so bad that the pipe must be dealt with from the outside, dug up and repaired. This is a most extreme condition and most cases do not involve quite that level of action.