Answer: Based on your description, the tree is undergoing some severe transplant shock. This is not really normal and the roots along the trunk are a sign of a poorly cared for tree.
It is very important to keep the root ball and the surrounding native soil moist while the tree becomes established. Willows are water lovers, so this is particularly important especially if the tree has been stressed, which it apparently has been.
When you water, soak the potting soil thoroughly and make sure the surrounding native soil is also moistened. (Roots will only grow into moist soil, so to encourage them to spread into the native soil it also needs to be moist.)Check often to make sure the potting soil is staying hydrated. Often, a quick draining soilless mix is used and this can dry out faster than the surrounding soil. The tree should be watered deeply to encourage deep roots. It is best to water deeply and less often rather than a daily light sprinkling.
Using a layer of mulch several inches thick and wider than the root zone helps conserve moisture and hold down weeds. Do not pile it against the bark however.
Willows have an uncanny ability to sprout either roots or shoots along the trunk and even from branches after the tree has been cut down! I would suggest gently rubbing off any sprouts or roots coming from the main trunk below the typical branching at the top. This will help direct energy to the top normal branches of the tree.
Losing those supplemental roots (they were pulling water from inside the plastic, possibly from condensation or overhead watering) may be adding to the water stress already caused due to transplanting. The bark may also have changed texture and may be undergoing stress due to exposure to the sun.
When severely water stressed, a tree will drop foliage as part of its efforts to compensate. Since it is late in the growing season this will not necessarily harm the tree in the long run, although it does mean that it will be producing less energy for the time being. Note that fertilizing at this time is not a good idea.
Keep watering as needed until the ground freezes. This is important in helping the tree come through winter with a minimum of die back. In the spring, trim away any branches that may have been killed off during the winter. Next season, keep watering while the tree continues to become established.
Good luck with your tree!
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