A Shih Tzu (English: ; ), also known as the Chrysanthemum Dog, is a toy dog breed, weighing 10–19 pounds (4.5–8.6kg) when fully grown.
 The exact origins of the breed are unknown, but it is thought to have originated in Tibet and then been developed in China.
The Hamiltonstövare is a breed of dog, bred as a hunting hound.
 The breed was developed in Sweden by the founder of the Swedish Kennel Club, Count Adolf Hamilton.
 Its ancestry includes several German hounds as well as English Foxhounds and Harriers.
Gobi is a small, female border terrier mix dog that began following ultra marathon runner Dion Leonard at the beginning of Stage II of the Gobi Desert Run in China in June 2016.
 Some vets have now told Dion that the dog could be a mix of Shih Tzu, Pug and Chihuahua.
Mal-shi, also known as Malshi, Malt-tzu, Maltzu, and Malti-zu, are a type of designer dog resulting from a cross between a purebred Maltese and a purebred Shih Tzu.
 Most Mal-shi are F1 crosses, as there has been little interest in developing the dog as a breed.
 They are not recognized by any major kennel club.
The Tibetan Spaniel is a breed of assertive, small, intelligent dogs originating over 2,500 years ago in the Himalayan mountains of Tibet.
 They share ancestry with the Pekingese, Japanese Chin, Shih Tzu, Lhasa Apso, Tibetan Terrier and Pug.
The Chinese Imperial Dog is a Toy dog breed with a wrinkly, short-muzzled face, and curled tail.
 Kennel clubs originally classified the breed under the shih-tzu, before recognizing the Imperial as a separate breed.
 Other names for the Chinese Imperial Dog include: Imperial Shih Tzu, Lion Dog, Micro Shih Tzu, Miniature Shih Tzu, Princess-type Shih Tzu, Tiny Teacup Shih Tzu, Tiny Toy Shih Tzu.
Zuchon or Shichon is a hybrid breed dog or a designer dog.
 It is cross between Bichon Frisé and Shih Tzu.
The Pekingese (also known as the Lion Dog, Peking Lion Dog, Pelchie Dog, or Peke) is an ancient breed of toy dog, originating in China.
 They are called Lion Dogs due to their resemblance to Chinese guardian lions (the Shih Tzu is also known as a Lion Dog in Chinese).
The Shih-poo is a small domestic dog.
 It is a crossbreed between a Poodle and a Shih Tzu.
 The name "Shih-poo" is a portmanteau of the two breed names.
Russian Tsvetnaya Bolonka (Russian: Болонка ), also known as the Bolonka Zwetna in Germany, is a rare toy breed of the Bichon type, developed in Moscow and Saint Petersburg, Russia from the ancestors of smaller dogs such as the Bichon, Toy Poodle, Shih Tzu, Pekingese and French Bolognese.
 They include the white version Franzuskaya Bolonka, which is a variation of the Italian Bolognese dog.
 "Franzuskaya" means French, "Zwetna" is the German term for "Tsvetnaya" and means multicolored, and "Bolonka" translates as "Bolognese" in a number of Slavic languages.
 The Bolonka has recently gained more fame by being the chosen breed of Prince William and Kate Middleton.

Question: Does the Hamiltonstövare or the Shih Tzu or both have a well established breed origin?
Answer: Hamiltonstövare