According to the encyclopedical manner (and what COBUILD advises too), we should put an article, be it definite or indefinite, before the name of a plant.
A rose is a plant… The coconut tree is a member of… etc.…
But when it comes to common names which include Saxon genitive, the problem seems to be complicating…
I do not consider the following webpage as a reliable reference, but they keep omitting adding articles to the plant called commonly as "Lady's Thumb".
https://www.farms.com/field-guide/weed-management/ladys-thumb.aspx
Please, folks, help, as I would also opt for no article, but, all the same, I would write the burro's tale. Lady's eardrops or the lady's eardrops?
What I am aiming at is to get rather an informative register than a colloquial or common one.