Paper birch (Betula papyrifera) are not native to this area and don't perform well, even though many landscapers have placed them in the environment. They  are primarily a Canadian species and their native range extends southward only through the northern US, where they can be found in mountainous regions or in other localities where a cool, moist site is available. Lewes, Delaware is definitely not a location to plant this species. Too hot, dry and the under soil is sandy. The tree is definitely stress trying to survive in this climate.

They become very susceptible to bronze birch borer attacks. Have you noticed any small holes in the bole of the trees (usually paper birch are planted in clumps of three)? Any small gatherings of sawdust at the root collar? This could be a sign of borer presence. Your tree may be also suffering from Birch dieback. Premature leaf drop is a sign that the vascular system has been compromised. Here's a good publication from the US Forest Service on the Bronze Birch Borer: http://na.fs.fed.us/spfo/pubs/fidls/bbb/bbb.htm

Look for signs of BBB damage. If the tree's decline is extensive and this insect is present, it is advisable to remove the specimen and replace with a 'native' tree. Birch prefer moist areas near stream (fresh water) banks and even though River birch (Betula nigra) is native to Delaware. It is not advisable to plant this specimen in your area of coastal Delaware. If you remove this Paper birch and want to replace the site with a tree, I would recommend one that is more tolerant to the surrounding climatic conditions, soil texture and water requirement. ~DOT

Thank you!

I thought it was a little strange to see birch trees this far south. Unfortunately, there are four birch trees on the property. Three are located on a hill near the drainage easement with plenty of sprinkler heads from the underground irrigation system near them. These trees are currently doing well. The tree that needs the most attention is near the patio so I think for now I will get some mulch to cool the roots. So far I do not see evidence of a pest infestation on any of these trees. 

One additional concern.... Any ideas as to what is causing the damage to the leaves?

Most likely, roots are dried-out and not able to produce or store needed nutrients for leaf development and retention. If insect borers are actively chewing inside the cambium, they are 'plugging' the vascular system. The planting location stress (a.k.a. 'environmental') may be also attributing to potential insect-feeding on the leaves. Without seeing the tree, it is hard to diagnose the exact problem. Supplimental irrigation during the summer months will not provide sufficient water needs to the Paper birch throughout the entire year. The planting site needs to consider the entire climate/environment during the life of the specimen (weeks, months, years). ~DOT