SHANGHAI, June 1 (Reuters) - China stocks eked out gains on Tuesday, helped by strength in healthcare firms, as investors cheered Beijing’s latest three-child policy after recent data showed a dramatic decline in births in the world’s most populous country. ** The blue-chip CSI300 index was up 0.2% at 5,341.68, while the Shanghai Composite Index added 0.3% to 3,624.71. ** Shenzhen’s start-up board ChiNext slipped 0.3%, while Shanghai’s tech-focused STAR50 index firmed 1.2%. ** Leading the gains, the CSI300 healthcare index climbed 1.7%. ** Birth- and fertility-related companies advanced as investors continued to cheer Beijing’s major policy support. ** Married Chinese couples may have up to three children, China announced on Monday. Beijing had scrapped its decades-old one-child policy in 2016, replacing it with a two-child limit to try and stave off risks to its economy from a rapidly aging population. ** Jiangsu Aoyang Health Industry Co Ltd, Shanghai Aiyingshi, Beingmate Co Ltd, Changchun High & New Technology Industries Group Inc and Goldlok Holdings Guangdong Co Ltd climbed between 5.2% and 10%. ** But analysts and traders expected limited impact from the policy shift on the market as a whole. ** “The three-child policy could have an impact on China’s long-term economic growth, while its stimulus could be limited for those who already have two children,” Luo Kun, an investment manager at Chasing Securities’ equities investment arm, said. ** Luo said he did not see a major impact on the stock market, noting gains for related companies could mainly be a result of short-term speculation. ** China’s market regulator fined 15 private tutoring firms a combined 36.5 million yuan ($5.73 million) for false advertising and pricing frauds, the official People’s Daily newspaper reported on Tuesday, amid a crackdown on the sector. (Reporting by Shanghai Newsroom, Editing by Sherry Jacob-Phillips) Our Standards: The Thomson Reuters Trust Principles.