Michael Justin Fitzgibbon, the 22-month-old Waldorf, Md., child who disappeared with his teen-age baby sitter early last month, flew home yesterday with his tired and teary parents. The plane ride from Dallas was the last leg of an odyssey that unfolded, according to police, as the child and 14-year-old Tammy Giles hitchhiked with random travelers and long-distance truckers across the country.

Police said the end came when Giles turned herself in at the urging of a friend who had just been released from a Pittsburgh jail. The teen-ager's memories of the whereabouts of the child were hazy. Michael, officials were told, had been left with a prostitute in a motel in Dallas. Police subsequently were able to track down the child.

"It's just incredible," said a red-eyed Sharon Fitzgibbon, 34, watching her son, in diapers and a white T-shirt, play with a plastic jack-o'-lantern in the living room of the Fort Washington home of friends, where she has stayed since the child disappeared. "I can't describe how I feel."

Michael, who had been missing since Aug. 5, appeared to be in excellent health yesterday. His curly blond hair was cut short, and he had a bad case of diaper rash, but his mother said he had gained weight, and spoke more coherently than he did before his disappearance.

The baby's father, Bob Fry, 43, who doesn't live with the family, became misty-eyed and hesitant during an interview. "I just figured the only way they'd ever find him was if she turned herself in," he said. "My hopes had dwindled from about 90 percent to 20 percent."

The search for the baby and baby sitter spanned several states and involved scores of law enforcement officials, including the Federal Bureau of Investigation. Many people, including long-distance truckers, telephoned to say they had seen the baby sitter and child on highways, or at truck stops, but the authorities were always several steps behind.

"Our information was always old," said Sgt. Dave Bowling of the Charles County (Md.) Sheriff's Department. "We kept showing up at places she'd been, as much as four weeks late. It was a problem we kept having -- nobody was saying 'I saw her five minutes ago.' "

The baby sitter, Tammy Giles, contacted police in Pittsburgh on Saturday, after her former boyfriend, who was released from jail there Friday, persuaded her to do so, according to Detective Joseph Tully of the Pittsburgh Police Department.

Giles, who arranged to be picked up at a downtown phone booth, is being held in a Pittsburgh detention center, police said. She is expected to be extradited early this week to Maryland, where she has been charged as a juvenile with kidnaping and abduction.

"The girl is a mess," said Capt. Therese Rocco of the Pittsburgh police. "She gave us vague information about where the baby was. She couldn't remember the name of the motel, but she gave us descriptions."

According to police and family members, the incident began after Sharon Fitzgibbon, who was going away for the weekend, left Michael in the care of his 15-year-old sister, Patty, on Aug. 4.

"I had left him before," she said yesterday. "My daughter is very, very responsible. This is a problem I face, like every other single parent."

When Patty wanted to spend the night with a friend, Giles, a school acquaintance of hers, volunteered to take care of the baby, Fitzgibbon said. She said Giles had visited the home only once before.

"I wouldn't know her if I fell over her," said Fitzgibbon.

Patty played with Michael at the house yesterday, but didn't want to talk about the matter.

Police said that sometime late Aug. 4, or early the next morning, Giles left Fitzgibbon's Waldorf home with the child. They believe she hitchhiked a ride to the Rte. 5-Branch Avenue exit of the Capital Beltway.

Described as a short, blond girl with round cheeks and a confident air, Giles had run away from home 22 times in the last 1 1/2 years, traveling as far away as Tennessee, Texas and California, authorities said.

Rocco described her as "a very street-wise girl who looks a lot older than 14." The night of the child's disappearance, Giles wore heavy makeup, jeans and running shoes; the baby was in pajamas and a sleeping bag blanket, according to motorists who gave her a ride.

About 1 a.m. on Aug. 5, Giles was picked up on the Beltway by a middle-aged man in a car, who drove her to Rte. 270 and Montrose Road in Montgomery County. The man, whom police would not identify, told authorities Giles had only $16 with her, and that he gave her $10 more.

About 2 a.m., she was picked up on Rte. 270 by Roma Campanaro and two friends, who were returning home to Montgomery County from Georgetown. "She said her husband had been beating her, and that she was going to see her mother in Pennsylvania," Campanaro said in an interview a few weeks ago.

Campanaro said she and the others decided to drive the young woman to her destination, Breezewood, Pa., a town two hours away, at the intersection of several highways.

The group offered to take Giles to her mother's home in Breezewood, but Giles refused, saying her mother would be angry if she knew Giles had been hitchhiking. So they took her and the child to the Breezewood bus station, where Giles said she would telephone her mother to pick her up.

The next report came from a truck driver who told police he picked up Giles at a Jessup, Md., truck stop. He said he was driving in tandem with another trucker, and they took Giles and the baby to Phoenix, where they left her at the Roadrunner Truck Stop on Aug. 12.

Police officials' next tip came from a truck driver, who said he had picked up Giles and the baby in Weatherford, Tex., just east of Dallas, and left her in Mesquite, on the other side of the city, on Aug. 14.

Believing her to be in Dallas, police and the FBI concentrated their search there.

Police say Giles apparently stayed at a string of motels in Dallas. On Sept. 8, she asked a woman at the Dallas Inn Motel, where Giles had been staying, to keep the baby for a few days, according to Bowling.

When Giles hadn't returned to the motel three days later, the woman, who Rocco said was identified by Giles as a prostitute, gave the baby to a couple living in the motel, police said.

Authorities, meanwhile, knew that Giles' former boyfriend, Mark Anthony Flick, was scheduled to be released from the State Regional Correctional Facility in Greensburg, Pa., on Friday.

"We hadn't had any recent sightings of Giles and the baby on the East Coast," said Sgt. Bowling, "but the FBI put him under surveillance, and then later talked to him."

On Saturday, Giles contacted Flick, and he persuaded her to turn herself in, Rocco said. Giles called and arranged a meeting at a phone booth not far from police headquarters, at Ninth and Liberty streets, according to Pittsburgh Detective Tully.

"She claims she arrived in Pittsburgh four days before she turned herself in," said Pittsburgh Capt. Rocco. "She wouldn't say where she was living. She said she just bummed around."

Alerted by Pittsburgh police, Dallas police located the child, and turned him over to a welfare agency.

Back in Maryland, Sharon Fitzgibbon spent Saturday afternoon helping some friends shop for a computer.

When she returned, the friends she had been staying with ran to the car to tell her that a child who fit Michael's description had been located in Dallas. "I couldn't even pack," Fitzgibbon said. "People had to do everything for me."

She flew to Dallas with Fry, arriving at about 2 a.m. yesterday, and they drove to the welfare agency immediately.

Before they could see Michael, they had to verify he was their child. They described his chipped tooth, his dimples, the way his left foot curves inward. Somebody at the home described the child as disobedient. "That's when I knew he was ours," Fry said. "He never takes 'no' for an answer."

The welfare workers woke Michael, and brought him out. "He was still half-asleep," Fry said, "and the side of his face was all wrinkled from the pillow. Then he got a little bit of a grin."

The trio then went to a hotel near the airport to await their morning flight back home. The baby went to sleep, but Fitzgibbon couldn't. "I just sat up and watched him, and cried," she said.