What you are seeing is the work of tiny, harmless insects called psocids or “tree cattle”. They feed on algae, lichen, and other bits of organic matter and pose no threat to you or the tree. At certain times of the year they spin these silk stocking-like webs which will disintegrate after a period of time. No action is necessary

Additional Resources

http://edis.ifas.ufl.edu/IN553

http://edis.ifas.ufl.edu/IN078