Pull them out. They are from the deciduous roots that the tree peony was grafted onto.

When it is done flowering, you may try air layering the tree peony to try and get it to grow roots from the tree peony section instead of letting it depend on energy of the deciduous roots because all that energy from photosynthesis is going right back down to the deciduous part.

This fall when your tree goes dormant I would dig it up and replant it deeper, so none of the rootstock was above the soil. Foliage growth from the rootstock is robbing your tree and will continue if not kept underground. When planting tree peonies I always plant at least four inches of the tree above the graft into the soil. This prevents the competition you are experiencing. Al

Thank you, I was worried that I would harm the root if I pulled them out. So remove them now and in the fall replant the tree at least 3 inches deeper....how far into the fall....late Sept, early Oct?

BTW both of my tree peonies are full of blooms

Thank you

No, don't dig them now. Dig in the fall and replant at the same time. Dig only after they have lost their leaves, or at least after all have turned color. Trees your size will be easy to dig. Al

Thank you Al, I meant remove the suckers now and replant the tree in the fall......both have bloomed beautifully this year....I had 10 flowers on the tree in the front yard but only 2 on the tree in the backyard....but that one had more suckers and is not planted as deep.

I will correct that for next year.

Thank you again