The Short Version
This claim is false — it gets the comparison backwards. NASA and ESA estimate the Milky Way contains roughly 100–400 billion stars, while a landmark 2015 Yale/Nature study estimates approximately 3 trillion trees on Earth. Even using the highest credible star estimates, trees outnumber Milky Way stars by a factor of roughly 7 to 30. The popular belief that stars vastly outnumber trees is a common misconception.