In one of Woody Allen's short stories, a shirt salesman falls upon hard times and prays for divine guidance. "Put an alligator over the pocket," God tells him. The puzzled man asks "Pardon me, Lord?" The reply: "Just do what I'm telling you—you won't be sorry:" Begging the Lord's indulgence, but René Lacoste got there first. The 75-year-old former French tennis star is the founder and reigning genius of his family's sprawling apparel empire—whose 1¼"-inch alligator trademark is recognized in more than 80 countries. In France it's called le crocodile—the fond nickname of the elegant sportsman (with prominent nose) who started the company 45 years ago.

Croc or gator, this year the little creature has been multiplying at a prodigious rate, bringing estimated worldwide sales for Lacoste to more than $350 million. In the U.S., which accounts for half those revenues, the scaly status symbol first turned up on weekend athletes like Dwight Eisenhower, Bob Hope and John Wayne. They began wearing the classic Lacoste shirt back in the 1950s. Now millions of Americans own something with the alligator emblem—shorts, shirts, sweaters, socks, pants, hats and belts, ranging in price from $3 to $90. Besides golf and tennis outfits, Lacoste also turns out clothes for boating, jogging, skiing and horseback riding.

Lacoste is even making a children's and infants' line. "Start those kids at 2," exults Vincent Draddy, chairman of David Crystal, Lacoste's U.S. licensee, "and they will still be wearing the alligator when they are 70."

The Lacoste empire was launched almost by chance. In the late 1920s René always played in the standard white pants and starched shirt with collar and cuffs. "One day I noticed my friend the Marquis of Cholmondeley wearing his polo shirt on the court," remembers René. " 'A practical idea,' I thought to myself." It was so practical, in fact, that René commissioned an English tailor to whip up a few shirts in both cotton and wool. "Soon everyone was wearing them," he smiles.

Why the alligator? On tour with the French Davis Cup team in Boston in 1923, René admired an alligator suitcase in a store window. He told his trainer that if he won he wanted the bag as a reward, and he was soon being referred to by friends as "Alligator." René lost the match and the prize—but the name stuck anyway.

He patented his shirt with the insignia in 1933 and went into business. Today the firm is still struggling to keep fakes off the market. In Japan it has been forced to stitch the family name across the alligator so there will be no doubt about the genuine article.

Lacoste retired from competition in 1929 after an attack of tuberculosis. Two years earlier he had defeated Bill Tilden for the American championship. Recovering from the illness, René married another athlete, Simone Thion de la Chaume, winner of the British women's golf title in 1927. (Their only daughter, Catherine, won the U.S. Women's Open 40 years later.) Nowadays the Lacostes spend as much time as possible at their Kennedy-like compound at Chantaco, the family's golf club in hilly southwestern France. Of Rent's three sons, Bernard, 48, president of Lacoste, is generally credited with its rapid growth. Michel, 36, joined the firm a year and a half ago. Francois, 46, is a Ph.D. in physics who works for a chemical company. There are 12 grandchildren. René continues to play tennis as a hobby and with his sons recently invented a racket that prevents tennis elbow. (Back in 1965 he designed Wilson's first metal racket, the T-2000, still a favorite of Jimmy Connors.)

On crisp fall afternoons at Chantaco, le crocodile enjoys lecturing the family on physical fitness—he's against wine, pepper and dancing—and sharpening his grandchildren's tennis game. "Perhaps," he muses, "there's another champion among them."