Bella's Lullaby Question?

Okay, so I love the song Bella's Lullaby in the Twilight series. What I love even more is the version of the song that was almost chosen tobe used but wasn't that everyone claims Robert Pattinson wrote. However, I just recently heard "River Flowing In You" by Yiruma, which was the same song. Check both on YouTube. So... what's the deal. Did he write the song or not.