"That branch caught fire," NTSB spokesman Chris O'Neil said.
O'Neil said all 13 people safely evacuated the balloon's basket, which then caught fire.
The group attempted to put out the blaze with fire extinguishers, but it consumed the basket.
The NTSB and the Federal Aviation Administration are both investigating.
It's not yet clear if the pilot was trying to land when the puncture occurred.
"The sequence [of events] and the mechanics will become clearer as we go on," O'Neil said. "It's fortunate that no one was hurt."
No injuries were reported and no other information was immediately available, FAA spokesman Ian Gregor said.
The hot air balloon is registered to Hot Air Expeditions in Phoenix. A phone message left with a Hot Air Expeditions employee was not immediately returned.
- AP