These names have the same meaning as the number of stars used for ranking cuisine at Japanese restaurants in the traditional style. Although this ranking system is rarely used by now, you might still find these names in the menu when you go to a restaurant, like Unagi, Sushi, or Kappou restaurant.
The ranking goes like this:
Rank 1: Matsu (pine tree) 松（まつ） - 5 star
Rank 2: Ta-ke (bamboo) 竹（たけ） - 4 star
Rank 3: Ume (plum tree) 梅（うめ） - 3 star
In most cases, the quality is basically the same for all of them, but the quantity or assortment of foods in dishes are different.
So, if you go to a Japanese restaurant, you might see Matsu Ta-ke and Ume in the menu and a waitress might ask, "Which do you like to have? you have 3 selections Matsu, Ta-ke and Ume."
In each case, you can select one that you like to order.
Well, even if you forget about Matsu, Ta-ke, Ume, you will find the difference by seeing the price. : ) Any ways, this would be of help when you go to a traditional Japanese restaurant in Japan.