According to SFGate Home Guides, published by the San Francisco Chronicle, a number of factors can contribute to the death of a weeping willow tree. First make sure that the weeping willow is planted in a USDA-approved plant hardiness zone where it can survive and thrive, which are zones 6 through 9. Willows require plenty of water to survive, which means a location near a lake, river or wetlands.
If the tree was recently transplanted, it is most likely a victim of transplant shock. SFGate Home Guides emphasizes leaving a recent transplant completely alone other than making sure it is amply watered. This includes not fertilizing the tree at this point.
Check for insects and also for holes in the trunk and branches of an established willow tree. Some of the common pests that invade weeping willows include scales, caterpillars, aphids and borers. The borers are generally reclusive, but they make holes in the tree's trunk and branches and cause its leaves to turn yellow and drop. SFGate Home Guides states that all of these insect pests can be controlled with use of the appropriate pesticides, but they need to be discovered early enough. If galls form near ground level on the trunk, the willow like is infected with crown gall, a fatal and contagious disease.