Q.weeping cherry blossoms
My weeping cherry only has blossoms on the ends of the lower branches. The blossoms only appear on the bottom fringe of the tree. It resembles a bald man’s head with no blossoms on the rest of the canopy.
Some other thoughts...
Is it possible your buds got nipped by frost?
Birds can also be a culprit - I have read where birds will eat the blossoms off a tree:
http://www.georgiabackyardnature.com/georgia_backyard_nature/2011/03/cherry-blossom-birds.html
Is your tree in a location with full sun? The more sun your tree receives, the more profusely it will bloom.
Has your cherry ever flowered well, or has this been the situation since you planted it?
How old is the tree? It is possible your tree isn't mature enough yet to reach its full blooming potential.
I have not been able to find anything in my research that would identify your description of you tree.
Have you inspected the plant carefully for any signs of insects or disease?
Please read the links I provided and perhaps identify if your tree is receiving enough water and fertilizer.
https://www.gardeningknowhow.com/edible/fruits/cherry/cherry-tree-pruning.htm
https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/cherry-tree-no-longer-weeps.htm
https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm