Image copyright Chutian Metropolis Daily Image caption Jenna Cook puts up a poster to try and find her birth family

When Jenna Cook went back to China at the age of 20 to search for her birth parents, she knew she was unlikely to succeed. What she didn't expect was that she would meet dozens of families who desperately hoped she was their lost child.

Near a busy bus station in the Chinese city of Wuhan, on 24 March 1992, someone left a baby to be found. It's quite likely that they watched and waited from a safe distance until the girl was spotted. She was picked up and taken to the Wuhan Children Welfare House, close by. There she was given a name, Xia Huasi, meaning "China's", and assigned a birth date chosen at random by the director of the home.

China's one-child policy meant that families faced heavy fines for having too many children. But it was also - and still is - illegal to give up unwanted children. There was no formal adoption process.

But just days later China passed a law allowing foreign nationals to adopt, and at the end of June an American primary school teacher, Margaret Cook, came to collect Xia Huasi. She renamed her Jenna and took her home to Massachusetts.

Jenna was one of the first wave of about 200 Chinese babies to go to American families. Many others followed - an estimated 80,000, mostly girls, have now gone to live in the US, and an additional 40,000 to the Netherlands, Spain and the UK.

Jenna always knew that she was adopted. "We would talk about adoption just like we would talk about what's for dinner. It never felt like something that was a big deal," she says.

Nevertheless, she sometimes wondered where she came from.

"Even just looking at your own belly button, you think to yourself: 'Oh, I used to be attached to another human being. That's the body I came from, but who is that? Does that person even really exist?' It all seems so abstract. It sometimes just feels like you appeared on the planet.

"Most people are just born into the families they're born into and they never think twice about it. Whereas for adopted people there is always this possibility of another life."

Image copyright Chutian Metropolis Daily Image caption Jenna and her adoptive mother, Margaret Cook

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Jenna and her sister, who was also adopted from China, grew up in an area where very few people looked like them. Their mother, Margaret, did what she could to maintain a connection to her daughters' country of origin - the girls learned Mandarin at school and they socialised with other families like theirs.

When Jenna was a teenager she was one of four Chinese adoptees to feature in the acclaimed 2011 documentary, Somewhere Between. Director Linda Goldstein Knowlton had adopted a baby from China herself and wanted to document the lives of these young women - drawing the title from something Jenna said: "I don't think that I could ever consider myself fully Chinese or fully American - I'm always going to be sort of somewhere between."

The 15-year-old Jenna captured in the film is a hard-working, high-achieving A-grade student. She is successful and loved, but haunted by a nagging doubt. Why did her parents give her up? Had she done something wrong? It's partly what drives her to be a perfectionist. In a moving moment in the film, Jenna speaks at an event for prospective adoptive parents and breaks down when she is asked how she feels about the word "abandoned".

"There's definitely a part of me that wishes I'd never heard the word 'abandonment'," she says.

Over the course of the film, Jenna delves deeper into her past and ends up volunteering for summer work at the very Chinese orphanage that took her in as a baby.

Image copyright Chutian Metropolis Daily Image caption Jenna Cook visits the orphanage where she stayed in 1992

Not all of the participants in Somewhere Between have a desire to return to China or find their birth families. In any case, adoptees are warned that attempts to trace birth families are unlikely to succeed. There is often very little information available, as birth families had to hide their identities for fear of punishment. And the records that existed in the 90s, when international adoption began, were badly kept. Add to this the sheer size and population of the country and it is a daunting task.

But miracles do happen. Haley, one of the four girls featured in Somewhere Between, goes back to the village where she was found. While putting up posters she is recognised by a woman who immediately runs to fetch her family. The next thing Haley knows, she is being hugged and kissed by a man who says he is her father. An emotional family reunion follows in which Haley meets her mother and sisters - surprisingly, she has more than one. Everyone looks shell-shocked by the experience.

In the West, adoptees searching for their birth parents can usually afford to take things slowly. But international adoptees don't have that luxury. They can probably only afford one such trip in their lives, says Bea Evans from the specialist company, Adoptive Family Travel. For more than 20 years she and her colleagues have taken internationally adopted children and their families to 18 countries including China, Guatemala, India and Korea (the Korean War led to the first wave of international adoptees in the US).

"Almost all international adoption has started in response to some kind of upheaval, whether it was political or financial or a policy like China's one-child policy," she says.

The company organises visits to orphanages - or social welfare institutes, as they are known in China - and occasionally assists with family searches and reunions. Evans says there is an increasing amount of "search and reunion" taking place in South Korea. Could this also take off in China? "I do wonder what will happen as more and more young [Chinese] women get to that age where they are saying: 'We want more information,'" she says.

The documentary maker Changfu Chang, who specialises in Chinese adoption stories, hears about successful searches almost every month. So how does he explain it? "Chinese society is a connected society, you do not really have many secrets," he says. "As long as you get into that particular village or neighbourhood or community others will help to provide that information."

But Jenna was found near Hongji long-distance bus station, where 12,000 travellers arrive in Wuhan from all over the countryside every day. This made the search particularly challenging.

Image copyright Getty Images Image caption A busy day at the bus station in Wuhan

When she was 20 and studying at Yale University, Jenna was given a grant to travel to China to begin her own search. It was partly an academic exercise - she hoped her experience could help some of her fellow 80,000 Chinese adoptees in the US. But of course it was also deeply personal, and she asked her adoptive mother, Margaret, to accompany her.

Jenna had printed flyers with pictures of herself at different ages and what little she knew about the circumstances in which she was found. She began handing them out to people in the streets of Wuhan, many of whom shared their own experiences. "Oh, I had a neighbour once who had a daughter in a similar situation," they told her. Or "I had a cousin who once gave up their child but I don't remember if it was in '92 or '93."

Image copyright Jenna Cook

Jenna found this fascinating. "I was pretty amazed that people were even paying attention to me, because I felt like I'm just one story in a huge migration of children from China," she says. "I felt like I was just one raindrop in the puddle."

But people were interested in her story, and a week after she arrived an article about her search appeared in the local paper. It was short and tucked away on page five, but the headline tugged at the heartstrings: "Dad, Mom: I really hope that I can give you a hug. Thank you for bringing me into this world."

It had a huge impact. In the weeks following the publication of that article on 25 May 2012, Jenna's search went viral. Hundreds of messages started coming in via social media.

"Their reactions were really polarised," says Jenna. Some people said: "This is fantastic that you're searching and I hope that you're able to find your parents and that your dream comes true." Others would say things like: "This is such a big mistake, you're wasting your time and energy." And: "You're so ungrateful to your American family, you need to go back to America right away."

Among the tide of messages there were genuine responses from people who thought they might be Jenna's parents. She narrowed it down to 50 birth families, each of which had left a baby on the same street in Wuhan in March 1992.

The implications of this are vast, says Jenna. What about other streets in the same month? What about other months? What about other years? What about the families who chose not to come forward? When she spoke to people who had worked at the bus station at the time, they said babies had often been left there.

But as well as being shocked by the sheer numbers, Jenna was surprised they were willing to come forward. After all, it is against the law to abandon a child - and after the publication of the newspaper article Chinese television had started filming her search. "Here are these people who have technically committed a crime and they're willing to come forward on national television. It was just unthinkable," she says.

Image copyright Jenna Cook Image caption The article about Jenna in the Chutian Metropolis Daily

Jenna and her mother arranged to meet the 50 families they thought could be a match. Some mothers and fathers came alone, but others brought the entire family, including grandparents. What surprised Jenna was that, far from being one-child families, often they had more than one daughter. What tended to happen was that they would keep their first daughter, and try again for a son. With each child they would incur penalties. Eventually, after having several daughters, they would decide to give one up, in the hope of saving a place in their family for a son.

Jenna initially approached the meetings from an academic standpoint. She told herself she was there to collect stories. "If I had gone into every meeting thinking: 'Maybe this is the one,' I would have been totally exhausted by the end of the day," she says.

But she still had to steel herself. "Especially for the first few meetings I was really nervous," says Jenna. "I was really worried about what they would think of me. I was really worried that maybe I had done something wrong, and that was why they had abandoned me - I worried that they would be angry at me."

Jenna thinks this is because she had unconsciously absorbed some of the prejudices that surround the issue. "In the US there is this dominant narrative that the reason why Chinese parents abandon children is because they don't like girls, and maybe they don't even remember them," she says.

But she found this not to be the case at all. "They all remembered their babies forever - it was this experience that they really regret and that they would never forget."

Image copyright Chutian Metropolis Daily

One woman brought a piece of delicate red-and-blue cloth that she had carefully kept - it was the material she had made her baby's suit out of. "She had kept these scraps for 20 years like a memory of her daughter. And she always dreamed that when they would meet her daughter would have the clothes and she would have the scraps - kind of like a lock and key."

Sadly, Jenna did not recognise the material. "I just remember shaking my head, I had never seen it. And the poor mother just collapsed. She was so devastated."

Another man she met, a long-distance bus driver, had spent a lot of time searching for his daughter. Whenever his bus route took him into the city he would go back to the area where they had left their baby and ask for her. They had left her with a note so she would grow up knowing her name.

Each family approached Jenna as if she were their daughter. For a brief moment, they represented the other's missing part. One mother even began brushing Jenna's hair. Mostly, they wanted to know if she was OK - like people emerging from a disaster and wondering if the other side had also survived, says Jenna.

They would ask: "Is your adopted mother good to you or does she hurt you? Does she give you enough food to eat?" Jenna would reassure them that she was well looked-after. "They would just be so happy to know that I hadn't been suffering all this time."

In turn, she asked them: "Was it something about me that made you relinquish me long ago? If I had been more beautiful or if I had been more obedient and had cried less would that have changed your decision?" And they were able to reassure her. "The parents just remembered their baby girl in such a loving way," says Jenna.

But there was also the business of verification. Having established that the facts matched, they would look for a physical resemblance - things like height, or foot-shape or hand-shape. Sometimes they would want to check for birth marks. Then, if they felt that there were enough similarities, they would go ahead with a DNA sample. In the end, 37 families opted to do DNA.

Sadly, all of the tests came back negative. It was a real blow.

"I think another reason why it was hard seeing all of the negative DNA results come back was because I sure wished I could be the daughter to every one of those families," says Jenna.

"To be the person that could help relieve their suffering - who wouldn't want to be that person?"

Despite this, Jenna feels the experience has helped her.

"Before, there was always a small part of me that felt like there was something I could have done 20 years ago to have changed my fate and then I wouldn't have been relinquished by my family," she says. "But after meeting the birth parents I realised it was really out of my control."

As an academic, it has changed her outlook completely. "It's a totally different experience to read in a history textbook about the one-child policy and read that parents abandoned their children or committed infanticide," she says. "But to meet people who have really lived that experience, and to see their great regret, and their great love for this baby - it's just something that's indescribable."

Jenna spent last summer working in China, but is no longer actively searching.

"I would love to have the chance to reunite with my birth family someday," she says. "But I can't say that will happen."

Jenna Cook appeared on Outlook on the BBC World Service. Listen again to the interview on iPlayer or get the Outlook podcast