The small, sparse foliage plus the excessive production of seeds suggest that the trees are under stress. Plants will often reproduce excessively when they are about to die. My references indicate that birch trees need well-drained soils with “ample water at all times and a regular feeding program.” They have a “poor tolerance of drought.” The 8 gallons of water per week may be insufficient for your trees if they are in a well-drained, sandy soil. The trees may also be nutrient-deficient. Small applications of a fertilizer high in nitrogen may help make the trees more vegetative and reduce their reproductive tendencies. I recommend that shade trees get approximately ½ pound of actual nitrogen per inch of trunk diameter per year. This amount should be divided into 3-4 applications spread over the spring, summer, and fall. Is there a salt or soil pH problem in your yard? Some areas may have soils high in salts and may be too acid or alkaline for good plant growth.