None of the injuries was considered life-threatening, though several people were trapped and had to be extricated by firefighters and paramedics, the State Highway Patrol said in a statement.
Patrol Sgt. Pete Combs said an investigation was underway and he could not comment on a possible cause for the crash, which happened around 4 a.m. on Interstate 75 in Liberty Township, about 25 miles north of Cincinnati.
But passenger Christopher Link, of Michigan, told WCPO-TV in Cincinnati that he saw the driver slumped over. Link said he heard a woman scream at the driver "Wake up! Wake up!" but that he thinks the man might have had a medical problem.
Link told the station the bus rolled over at least twice after hitting the tree.
Jeff Galloway, director of the Butler County Emergency Management Agency, said 35 people were taken to hospitals, six by helicopters and 29 by ambulance. The injuries ranged from minor to severe, officials said.
The bus, which left Cincinnati bound for Detroit, was carrying 51 passengers and the driver. Those passengers who were not injured and those who were treated and released from hospitals were transported back to Cincinnati, but none of them were at the bus station later Saturday morning.
At least nine people remained in hospitals Saturday night.
Eleven of 13 people taken to Atrium Medical Center in Middletown and all but one of the 14 taken to West Chester Hospital had been released as of Saturday evening, according to officials at both hospitals. The conditions of those who remained hospitalized were not released.
One person was admitted to Mercy Health-Fairfield Hospital in stable condition, according to spokeswoman Lauryn Moore.
Two patients taken to Bethesda North in Cincinnati remained in fair condition there Saturday evening, hospital spokesman Michael Mattingly said. Terri Ann Fredette, a spokeswoman for University of Cincinnati Medical Center, said two people were taken there. One was released and the other was in stable condition.
Two of the four people taken by helicopter to Miami Valley Hospital in Dayton were released and one was taken to surgery for a broken leg, spokeswoman Nancy Thickel said. The fourth was admitted for contusions and bruising.
Link, who was not injured, told WCPO he saw some children on the bus and thought some passengers had broken arms and legs. He said that the bus driver seemed fine when he boarded the bus.
The driver, who has been with the company for almost 15 years, had been on duty for an hour and was fully rested, Kim Plaskett, a spokeswoman for Dallas-based Greyhound Lines Inc., told the Associated Press.
The driver was among the injured, but she said she could not release the person's name or medical condition due to medical privacy laws.
Plaskett said she couldn't discuss any details of the crash or the possible cause. The company was cooperating with investigators and will talk to the driver and conduct an internal investigation to try to determine what happened, Plaskett said.
The bus just had its regular major annual inspection 14 days ago, Plaskett said. She said drivers also do pre-trip inspections to make sure buses are fit for travel.
Greyhound sent a crisis-response team to the site to help the customers and authorities as soon as the company was notified of the crash, she said.
A telephone hotline was set up for friends and family members seeking information about the passengers on the bus. The phone number is 800-972-4583.
The skies were clear in the region early Saturday morning, according to the National Weather Service.