Do names of plants and trees have to be capitalized?

Yes, the names of games are capitalized because the name of a thing is a proper noun; for example:

• Monopoly
• minecraft
• Candyland
• Grand Theft Auto
• Operation

Another example, The Games of the XXVII Olympiad is the offcial title which is usually shortened to the 2000 Summer Olympic Games. Both of these are proper nouns and both are capitalized.

However, 2000 summer games is not a title, it's an informal term and a common noun, which is not capitalized.