Weathering can occur either mechanically or chemically. Mechanical weathering occurs when larger rocks are broken down into smaller rocks. Their chemical composition is the same, but they are in smaller pieces. Chemical weathering breaks down rock by chemical reactions that occur with the minerals that make up the rock. This can occur with something as simple as water dissolving any salt minerals that are in rocks to acid rain reacting with limestone and marble and wearing them away.
Your question about tree roots is an example of mechanical weathering. As the tree roots begin to grow in the cracks of the rock, the roots will grow in diameter. This growth will slowly make the cracks in the rock larger, and eventually the roots may split the rock into pieces. Other processes may then act on the pieces of the rock, but since the tree roots played a role in breaking the rock apart, it is an example of mechanical weathering.