Pink, 31, recently spoke to PEOPLE about the meaning behind her newborn baby daughter's name, Willow Sage, who is the first child for her a motocross-riding husband, Carey Hart (35). During the interview, the 'Funhouse' singer also talked about the actual birth, and how her hopes of experiencing a natural birth were trumped during the delivery.

"The willow is my favorite tree. I grew up near one. It’s the most flexible tree in nature and nothing can break it -- no wind, no elements," she said of baby Willow's name. "It can bend and withstand anything. I love that sentiment. I want that for her."

She also spoke about her daughter's middle name, explaining, "Sage is cleansing and sacred. And it sounded great together. It doesn’t hurt that her last name is Hart -- flexible cleansed heart."

The new mommy also discussed her hopes of delivering Willow without any meds or surgical procedures. "We watched Ricki Lake‘s documentary 'The Business of Being Born' early on. It was so enlightening and empowering," Pink said. "We took all kinds of classes and decided to work with midwives at The Sanctuary … I was really looking forward to the whole rite of passage -- giving birth perfectly present, unmedicated, in the way nature intended."

However, Willow Sage wasn't as cooperative as her mother had wished. "She was in the frank breech position, which is head up with her legs up by her head in a pike position. We tried everything to turn her around," she continued. "Turns out this little girl had other plans -- she is my daughter, after all."

Despite having to undergo a C-section, Pink didn't mind since her concerns quickly turned to whether or not her baby would be OK. "It all turned out perfectly in the end, even though it wasn’t what we intended, because she is healthy and happy and so am I,” she explained, before hinting that Willow won't be the couple's only child. "We’ll go for our natural birth next time, for sure."

See a picture of baby Willow below!

Watch the Pink 'Raise Your Glass' Video