Growing up, my best friend had a tire swing in his backyard. While it provided trivial entertainment, looking back, a tire swing involves quite a bit of physics. First of all, there is the connection to the tree. The swing needs to be far enough away from the base of the tree in order to prevent accidental collisions, but it also needs to be sturdy enough to withstand human weight. The further out from the base of the tree you put the swing, the more torque that is applied to the branch as the lever arm is being increased. Perhaps this is why they have the "do not try this at home" warning on most innocent projects. Another aspect of the tire swing is the ideal motion of the swing, similar to that of a pendulum. Like a pendulum, the rope is nearly mass-less in comparison to the tire and human object, allowing for simple harmonic motion. However, air resistance and other movements prevent the pendulum from reaching its previous height each time, creating a dampening effect, even though it is extremely slow. In addition to this motion, the tire swing often rotates. Similar to the common figure skater example, a stretched out human will rotate slower than a balled up human, due to the indirect relationship between rotational inertia and angular velocity as they are multiplied to equal the consistent angular momentum. Finally, the tire swing usually ends up following an ovular or circular path instead of a straight line, so it would be interesting to study the exact physics of a pendulum in a non-linear path.
It is also important to remember to hold firmly onto the swing, as a fall from the swing could result in serious injuries (Witnessed it firsthand) due to the acceleration of gravity, but also unforgiving nature of the ground, specifically tree roots. The hard surface of the roots keeps a low impact time, causing a high force to be felt by the person as their momentum is immediately changed to 0.
EDIT: Didn't see Kate's blog on this, sorry kate :/