If you have very large tree almost nothing will grow. Seeds will start and germinate but with dogs trampling on it it will wear down to dirt. One of the main problems is the soil gets very compacted and if tree roots are along the surface, it will be a losing battle. Otherwise, if possible, rent a rotor tiller and brake up the ground about 4-6" if there are little or no root problems. Rake smooth and add some top, then seed. If possible, thin the tree out, not just cut the branches back but remove the entire branch. You want some sun light to penetrate to the ground. Tree in the spring time drop a lot of seeds and in the fall drop leaves. This material smothers the grass so you need to keep it raked and clear for the best results. It's hard for a definitive answer since we don't know the type of trees, yard location, how much shade and water, etc.