The suffix -ivore means eater. While herbi- means herbs or plants. -Omni means all. A herbivore eats only plants and an omnivore eats anything. Humans are omnivores since we can and do eating anything. Plants like sunflowers don't eat anything. They make their own food using sunlight.
You really should learn the meaning of words like these since you come across them over and over again. They actually tell you what they mean.
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The sunflower is an annual plant in the family Asteraceaeis. There are several varieties. The sunflower is a single stalked flower that produces seeds. The seeds are much used for feeding parrots, wild birds, and various pets such as rats, mice and hamsters, etc., as part of a mixed diet. The seeds can be used in cooking and will be on sale in health stores, either as seeds or as an oil. So, in a way, it is a kind of herb in cooking, but definitely not a shrub or a tree.