Next spring, spray the plant with a fungicide that is labeled for Botrytis every week from the time the new growth starts coming out of the ground until they bloom. Remove all dead plant parts now and again in the fall. The fungus can form little black spore pellets that can lay dormant in the soil for years waiting for the wet, cool conditions it prefers.
Here are some other tips for growing peonies. While peony clumps can last for generations, they may flower better if divided every decade or so. Do the division in the fall. The rhizome can be cut into divisions of 3 to 5 eyes, although small divisions may take a couple of years to begin blooming. Plant them with the eyes of the tuber only about two inches below the soil surface. Fertilize them each spring with a 10-10-10 or a 2-3-1 ratio fertilizer.
If you want cut peony flowers, only cut off about one-third of the flowers from each clump. Leave at least three or four leaves on each stem that is cut. After blooming, you can pinch off the seed heads. They grow and bloom best in full sun and in well-drained soil high in organic matter with a pH between 6.0 and 7.5. Water them when the top few inches of soil dries out, and water so the soil gets wet to about two feet deep.
It is good to plant trees while they’re dormant. The problem with buying a tree while it is dormant is that there may be some dead branches that you can’t know about until the tree leaves out. In your case, it sounds like the trees did begin to leaf out just fine.
There are many possible causes for a branch to die, but in this case, the topmost branches were probably killed by the nursery crew. I have seen many trees riding down the road on a truck or trailer without being protected from the wind. The tree may be mostly covered with a tarp, but the top end of the tree is sticking out and the leaves are flapping like a flag in a hurricane. Tree leaves can’t survive the drying wind and destructiveness of the wind if they are on the road very long. New leaves on a tree have the strength of wet tissue paper and are very susceptible to this type of injury.
Hopefully, the buds for more leaves were not killed and the original trunk will continue to grow straight up. If the top of the tree is dead for several feet, a new leader will have to be created from a branch and pruned into place to replace it.
I think the same thing happened to the tree that is dying on one side. It was the one that faced into the wind on the truck to your house. Again, it stands a good chance of surviving; you will just have to wait and see.