Thank you for reaching out with your question and providing photos of your birch tree so we can help figure out what is going on. Sometimes trees can be delayed in leafing out. It is promising that even the branches that do not have leaves yet do seem to have buds on them. 

One way to determine whether branches are alive is to do what is called a scratch test to see whether the layer under the outer side of the bark is green. You can also pinch a few of the buds on various parts of the tree to see whether they are green on the inside. If the tree buds are black or shriveled, then there may be other issues affecting the tree health such as nutrient deficiencies, compacted soil, pests, or restricted roots. 

Another thing to consider is that while river birches do tolerate wet conditions, you want to be careful not to overwater them. What is the drainage like around the birch?

Watering them every 2-4 days is generally sufficient this time of year (less if it's raining frequently) and 3-4 times a week during the hotter summer months. The roots need oxygen and water and if it's too wet, then you can inadvertently deprove the roots of sufficient oxygen. You can find more guidance on watering trees and shrubs here: https://extension.umd.edu/resource/watering-trees-and-shrubs/arec.umd.edu/resource/watering-trees-and-shrubs