Hi and thanks for contacting Ask an Expert.
Peonies are gorgeous and they can live for years. They do not like wet feet, so drainage may be an issue with the excessive rain we have had this last month. They are also somewhat drought tolerant, do not overwater. The cold/hot weeks we had may have also affected it.  
They thrive in full sun, so if a tree has grown to shade it that shade tree may need to be pruned. They also like the soil pH to be around 7.0 or neutral. You can purchase a pH test kit at your local nursery and test the soil around the tree. Just follow the directions.  
Start fertilizing with fish emulsion around the drip line (the edge of the tips of the branches). Pour the fish emulsion solution (see directions) on the soil every two weeks. This is a great source of all nutrition especially nitrogen for the plant. Do this through September. This will give the plant extra nutrients for next spring.  You should also see better growth and color in the tree.
If the branches that are wilting continue to die, cut them back to at least 8 inches from where they are wilting. In other words, close to the trunk.  
Look for little holes in the bark. If you find them, it could be a borer insect. Stick a wire into the hole as far as possible and wiggle it, killing the larva that is in the hole. Fill the hole with clay or white glue.  
Both Neem oil and lime/sulfur fungicide sprays can be used for any fungus that may be on the leaves. Look for white powdery mold, black mold, or black spots. Both can be used, one the first month and the other the next month. Do not use lime/sulfur on very hot days or in a long hot weather forecast. The sulfur can burn the leaves. Please read the directions carefully and follow them. The sprays are for use if you feel the plant has a fungal disease.
Peony trees bloom on old growth so do not cut too many of the branches. When pruning, cut branches growing across others, and cut to open the center of the tree giving it more air movement. You don't have to cut too many branches to get air movement unless the plant is really overgrown.  Mulch the tree in the fall with leaves, straw or wood chips.
Here is a link to Rutgers University's advice on peony trees. Please contact us again if you have further questions.

http://www.rutgersln.com/nursery/maintenance-and-care-tips/growing-tree-peonies/