Why pull the tree down anyway? If you like it, keep it and just replace the drain. Those old terra cotta pipes were flawd from the start, terra cotta lets water seap through and actively attracts roots towards it, which then later invade the pipe and wreck it. They also have lots of joins with gaps that let the roots in. It should also be mentioned waste water is full of nitrogen which roots are eager to get at.

Modern plastic pipes are the best in this case, they don't let water seap through, so the few roots that grow towards it will pass over it. And, should an expanding root shift the pipe it's flexible enough to move slowly with the roots and not snap or become lose at the joints. However, it's best to keep it clear of large plants and route it under a lawn or driveway where possible - but it will survive almost all stray roots easily.