No that's not normal. Cut off the dead parts. If it was just planted they must have sent you a half dead tree. Either that or your growing conditions are bad.

It does look like there are some good buds so hopefully the tree will take off.

make sure you have moisture down in the rootzone. I would prune off all the dead wood...back to a leaf or bud. Young cherry trees are tricky. My little Minnie Royal is not growing...

Those limbs were green when I planted and those buds weren't there. I'm in. 9a, east Texas, so it's very hot and wet all the time. We have two seasons here, and both are hot. The good thing is we have ample rain. I'm gonna go ahead and trim the dead wood, but should I leave the new leaves on the parts that are still green?

You can't grow a tree without leaves. Why would you remove them? And I hate to tell you this but wet isn't good for cherries. They do much better in dry climates with irrigation. If your soil drains poorly that could be contributing to the poor condition of your tree. In fact wet soil will kill a cherry in short order. Moist is good, not wet.

I've planted them on mounds, so they're slightly above ground level. I only water once a week, as long as it isn't raining. I got these on a whim and will try my best to make sure they do well. If they don't though, I won't be heart broken. I understand the tree needs leaves to grow, but some of the new leaves are close to the dead parts.
I appreciate the prompt responses! I hope these ladies take off!