Q.Pruning a Birch Tree
I have a small (center trunk about 5 feet high) white bark birch tree in my front yard garden.
It has been there for over 12 years, however, it has a frail, sparsely droopy leaf appearance. We have never pruned the tree because our gardener has said that it is not good to prune Birch Trees. My thought is that if we trimmed off the very thin branches (no bark) that emerge from the one inch white bark limbs may make the tree more healthy. Is that correct and also is it o.k. to do that at this time in early March? I have done some research and have read that I could damage the tree if I cut off the very thin branches that emerge from the one inch branches. I also have a white bark Birch Tree in another part of the front yard that is about 18 to 20 feet tall. The thin branches that emerge from the one inch white bark limbs also look frail. The limbs & leaves in the summer droop down.

Late summer or early fall is the best time to prune your Birch Trees.
This article will help you with your pruning project.
https://www.gardeningknowhow.com/ornamental/trees/birch/cutting-back-birch-trees.htm