"We wish it to be known how grateful our family are for the huge response locally and across the world in our search for Catherine", they said.
Her body was found by a search team about 60ft (18m) from the top of the Indian Nose hiking trail on Monday.
Authorities in Guatemala launched a search for her, and her body has now been found.More news: Twitter reacts to Anthony Barr backing out of agreement with Jets
Catherine, from Witney, Oxfordshire, has been travelling since September previous year and was reported missing on March 5.
Her father added: 'Please come home, sweetheart'.
Last night the Lucie Blackman Trust, which helps the relatives of people missing overseas, said: "We are now able to confirm that the body found in the search in Guatemala is that of missing Catherine Shaw".
Foreign Office advice cautions visitors that Guatemala has "one of the highest violent crime rates in Latin America", warning to "take care in all parts of the country". It's very sad what's happened but we're hopeful Catherine can be found'.More news: Raptors’ Kawhi Leonard reacts to Serge Ibaka-Marquese Chriss fight
"Catherine left the hotel on Tuesday March 5 in the early hours of the morning, at 1.37am to be exact, taking with her a musical instrument".
The area is famous for its lakes beneath towering volcanoes.
She set out travelling in September and had been with a friend in Guatemala for two weeks, having previously visited Mexico and California. Prior to checking into the hotel, she had been staying in a campervan in the San Pedro, Lake Atitlan area of Guatemala with a friend. Shaw's friend, Jess Elizabeth, wrote on a Facebook page dedicated to finding Shaw that the backpacker had "no belongings with her, passport, money or mobile phone - they were left at the hostel".
Police said in a statement Shaw had been missing since March 4 and that a criminal investigation would be opened into her death.More news: White House refuses to say whether Trump believes Democrats hate Jews
Calls to the family home, where Catherine's mother Anne and sister Jenny live, were referred to the Lucie Blackman Trust. We would ask anyone who thinks they may have seen her to contact us'.