Without the removal of the trees that is obtaining the water and nutrients and blocking the sun to those areas it is going to be difficult. You could trim the branches/leaves to allow more sunlight to those areas. We would also recommend trying to do your best to get nutrients to those areas in order for the new seed to thrive. You could apply a moss and weed killer to treat those areas prior to seeding (following product guidelines for timing).  You can get a soil test performed in your county through your local cooperative extension office to determine if lime is needed to adjust the pH and the deficiencies the lawn may have. We also have a cool season lawn care guide that may be helpful.