NEW YORK (Reuters) - Hillary Clinton promised on Monday to bring the "full weight of the law" against people who kill police officers if she becomes the next U.S. president after two recent episodes of gunmen slaying police officers in Dallas and Baton Rouge.

Clinton, the presumptive Democratic candidate ahead of a Nov. 8 election, made the promise in a speech in Cincinnati at the annual convention of the civil rights group the National Association for the Advancement of Colored People.

"Anyone who kills a police officer and anyone who helps must be held accountable," Clinton said. "As president, I will bring the full weight of the law to bear to make sure those who kill police officers are brought to justice."

The penalties for killing a police officer are rightly more severe than those for killing a civilian because police are symbols of the rule of law, she said.

Clinton's remarks come among heighten tensions between police forces and many black and Latino Americans, who have been rallying in protests across the country against a spate of police killings of black men, often unarmed, during encounters.

A black former soldier killed five police officers during one such protest in Dallas, Texas, on July 7. On Sunday, another black former U.S. Marine sergeant killed three police officers in Baton Rouge, Louisiana.

(Reporting by Jonathan Allen; Editing by Chizu Nomiyama and Alan Crosby)