I have a tree peony that sends up herbaceous shoots. They are easy to spot, once they leaf out (as noted by Liz). I cut and pulled them out last year. Waiting to see if they come back this year.
I think the herbaceous roots are supposed to die after the peony gets established. I believe they are called "nurse" roots. In other words, the tree peony actually develops it's own root system. The "nurse roots" are just used to help it along.
It's not like the normal "root stock" graft thats used for roses, trees, & conifers.
