Arborvitae in Trouble
The dying branches can be caused by dryness, physical damage or a fungal disease such as a Phytophthora root rot, phomopsis or other blight. It is difficult to tell what it is just based on the photos. The plants are close together, which can cause competition for water and nutrients and prevent air circulation. Keep the plants well watered during periods of drought. Air circulation is important to help prevent disease. Just general stress from competition can also make plants more susceptible to disease. I encourage you to get into the plant canopy and look for any symptoms of the diseases I listed, or insect damage from mites or scale. More info about what to look for here: http://ccenassau.org/resources/arborvitae-problems
For root rot, expose the roots on your arborvitae. They should be light-colored and not dark, mushy or friable. There may also be a discolored area at the base of the trunk of the tree from the fungus. Scrape away an area of bark, the area underneath should be light colored and firm, not dark and soft.
I would also examine the base of the tree and the branches that are dying (further back towards the trunk). Look for signs of gnawing damage caused by rabbit, chipmunk or other rodents.This gnawing can cause “girdling” of the plant or branches. Also look for any other signs of physical damage or possibly any signs of canker.
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