Damages occurring from blocked drains are a huge problem. Reports as far back as 2015 have shown that New Yorkers alone spent $18 million fixing blocked drains in just 5 years. And yes, the majority of these occur thanks to tree root ingress.
If you feel you have a blocked drain caused by a tree, the first thing you want to do is call a professional. They have a host of sophisticated appliances through which they carry out their analysis before even fixing the problem.
For instance, it may require a CCTV installation to identify the root of the problem (no pun intended). Afterwards, they can then make use of tools such as a mechanical auger or rooter. The auger gets fed into the pipe until it reaches the obstruction and from there it chops the root into bits.
Finally the chopped bits and pieces from the roots are then flushed out and your pipe becomes free again. I had a blocked drain in Medway caused by tree ingress and this was how it was handled.