Very interesting story. The swing vote was made by the youngest legislator in the TN house of representatives. He was against amendment, but changed his mind:
The next day, Burn defended his last-minute reversal in a speech to the assembly. For the first time, he publicly expressed his personal support of universal suffrage, declaring, I believe we had a moral and legal right to ratify. But he also made no secret of Miss Febbs influenceand her crucial role in the story of womens rights in the United States. I know that a mothers advice is always safest for her boy to follow, he explained, and my mother wanted me to vote for ratification.
Minutes after Tennessee ratified the 19th Amendment, essentially ending American womens decades-long quest for the right to vote, a young man with a red rose pinned to his lapel fled to the attic of the state capitol and camped out there until the maddening crowds downstairs dispersed. Some say he crept onto a third-floor ledge to escape an angry mob of anti-suffragist lawmakers threatening to rough him up.
The date was August 18, 1920, and the man was Harry Burn, a 24-year-old representative from East Tennessee who two years earlier had become the youngest member of the state legislature. The red rose signified his opposition to the proposed 19th Amendment to the U.S. Constitution, which stated that [t]he right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of sex. By the summer of 1920, 35 states had ratified the measure, bringing it one vote short of the required 36. In Tennessee, it had sailed through the Senate but stalled in the House of Representatives, prompting thousands of pro- and anti-suffrage activists to descend upon Nashville. If Burn and his colleagues voted in its favor, the 19th Amendment would pass the final hurdle on its way to adoption.
http://www.history.com/news/the-mother-who-saved-suffrage-passing-the-19th-amendment