Vladimir Putin once boasted to George W. Bush about the size of his dog, in the ultimate my "mine is bigger than yours" story.

Former President Bush writes about the episode in his memoir, "Decision Points," which hits bookstores on Tuesday. Bush says he had introduced then-President Putin to his Scottish terrier, Barney, on a visit to the U.S. presidential retreat, Camp David.

Putin returned the favor when Bush visited Russia and Putin was giving him a tour of the grounds of his dacha.

"A big black Labrador came charging across the lawn. With a twinkle in his eye, Vladimir said, 'Bigger, stronger, faster than Barney,'" Bush writes.

Bush says he later told the story to the Canadian prime minister, Stephen Harper, who replied, "You're lucky he only showed you his dog."

Bush, when he began his presidency in 2001, famously looked into Putin's eyes and said he saw his soul. "In the years ahead, Putin would give me reasons to revise my opinion," he writes.

Bush says the low point in his relationship with Putin came in August 2008 when Russia invaded Georgia. Bush complained to Putin directly at the Olympic opening ceremonies in Beijing and noted to him that Georgian President Mikheil Saakashvili is "hot-blooded."

"I'm hot-blooded too," Putin retorted, according to Bush.

"No, Vladimir, you're cold-blooded," Bush said he replied.