There's trouble in the White House Eden.

More than 2,000 garaniums in Lafayette Square, the 186-year old President's Park, have been eaten. More than half a dozen freshly-planted trees also have been gnawed to death, and the historic park's lawns and brick paths now are littered daily with chewed branches and fallen leaves from virtually every tree in the park.

The hungry villains are the common, apparently all too common, gray squirrels. They spent the past cold winter snuggling in their park tree nests and this spring produced the largest crop of ravenously hungry baby squirrels seen in the park in several decades.

"While I was out surveying the damage in the geranium beds - they ate over $2,200 worth of geraniums in two days - I heard this noise in a small tree beside me, a young beech," says Bill Ruback, National Park Service ranger in charge of the White House grounds. "And there right beside me were six baby squirrels chomping it to pieces . . . there must be 100 squirrels in that small square now, and they're just shredding the trees," says Ruback.

To temporarily keep the squirrels at bay, Ruback bought 200 pounds of raw peanuts, which Park Service employees now sprinkle enticingly at the base of the square's trees every morning (no, they're not Carter Georgia peanuts, they're Virginia peanuts). But squirrels cannot live on peanuts alone and the noisy crunching of branches still continues overhead.

Like Lafayette Square, almost all of Washington's downtown parks show the effects of large resident squirrel populations, although apparently few have had as big a baby boom this spring. The trees, even large 100-year-old oaks, are gnarled and slightly stunted from decades of almost continuous gnawing.

"Some trees look like dwarfs they've been chewed so much," says Ruback. The scars are evident everywhere, in lumpy cancerous-looking growths on branches, where squirrels eat the bark and the tree keeps trying to cover over the wound. When squirrels girdle or eat around the branch or tree, it dies. Thousands of small branches on Lafayette Square trees are withered testimony to this and more than half a dozen to 10-to 20-foot trees are now dead and leafless.

"What we want to do is trap the squirrels, a large number of them, and free them in parks and woods around Washington," Ruback says. "But the last time we tried that . . . in 1970 when the squirrels were bad, but nothing like this . . . the squirrel and animal lovers went wild and we got an awful lot of bad press over it. But we did reduce the number of squirrels."

President Eisenhower stirred an earlier but similar barrage of protests in 1955 when squirrels began digging up his recently installed White House putting green and Park Service employees trapped and deported them to nearby woodlands.

The traps were a last resort after an unpublicized electronic war against the squirrels was waged, and lost, around the putting green. Highpitched sounds wer beamed at them, but when the noise had no effect baited traps were used. The trapping was stopped abruptly however, after only three squirrels had been removed from their Presidential paradise, when investigative reporters uncovered the plot against the squirrels. Hundreds of newspaper editorials, columns, cartoons and letters to the President were written in defense of the arboreal rodent.

Former U.S. Ambassador to Great Britain Walter Annenberg had squirrels shot on the grounds of his residence in London several years ago when they began digging and eating his tulips, but this primitive American solution to the squirrel problem also was stopped after a hue and cry was raised in English newspapers.

"Actually, trapping and releasing squirrels in the country would be good for the squirrels as well as Lafayette Park or any overpopulated park," according to one of the nation's foremost squirrel experts, prof. Vagan Flyger, who heads the University of Maryland's center for environmental and estuarian studies.

Before any of Lafayette Square's squirrels are trapped, however, Ruback plans to file an environmental impact assessment, required under federal law for any "major federal action" affecting U.S. parks, which he hopes this eill answer the objections of squirrel lovers.

The population explosion at Lafayette Square may be heralding another great squirrel migration in the East, Flyger says, similar to that of 1968 when millions of squirrels from Maine to South Carolina crisscrossed the countryside in seemingly pruposeless journeys, like the celebrated mass migrations of their European cousins, the lemmings.

"We don't know exactly why it occurred but we think there was simply a limited food supply and just too many squirrels," says Flyger, "and the conditions this year are almost identical to 1968."

But squirrels have been doing this in North America for 55-60 million years and neither they nor their habits have changed much in that time, he says. A "Please Don't Eat the Geraniums and Trees" sign in Lafayette Square will not deter them. "The only thing to do is trap them," says Ruback. "But if we finally get permission to do it we'll have employees in civilian clothes trap them early in the morning. We don't want the public to see us doing it in uniform. We've had enough bad publicity."