Select the site for the weeping mulberry tree carefully. Mulberry trees have invasive roots that may damage sidewalks and patios. They also have dense leaves that prevent sod from growing underneath them.
Plant the tree in full sun, preferably in the spring while it's dormant. Weeping mulberries thrive in moist, fertile soil, although they tolerate dry, infertile soil as well. Water the tree weekly for the first summer until its roots become established. Water the tree during the fall and winter occasionally if conditions are dry.
Apply a wood chip mulch in a 6-inch circle around the weeping mulberry to conserve moisture and reduce weed growth. If you plant a weeping mulberry in a flower bed, select shade-loving plants and ground cover such as astilbe, ferns or wild ginger, which tolerate the shade produced by the weeping mulberry's dense canopy.
Fertilize the weeping mulberry with 1 pound ammonium sulfate 21-0-0 annually for each inch of trunk, as measured by the circumference at the bottom of the trunk. If the tree sits in or near a lawn that's regularly fertilized, don't fertilize the tree — it receives enough nutrients from the lawn fertilizer. Avoid using weed and feed products near the tree.
Prune the tree during the first two years to remove any branches that grow less than 6 inches apart. These narrow crotches are prone to splitting as the tree grows. Also, trim the branches back so they don't hang on the ground, if desired. Cut any suckers that spring up at the base of the tree back to the ground. Mulberry trees are prone to suckers, which are small, vegetative growths that sprout from the base of the tree.