TRAP a beaver, save a tree. That could be the new provincial motto of Manitoba, Canada, where beavers - 600,000 and rising - could soon outnumber people - 1 million and falling. For many Canadians, Canada's national symbol has become a national pest.Beavers are damming up streams and knocking down thousands of trees. Earlier this year, a flood from a beaver dam closed the Trans-Canada Highway, near Rainy River. Fields of wheat have been flooded. Prairie farmers plant poplar trees as a wind break; the beavers mow them down. Poplar trees are the filet mignon of the beaver world. "We beg trappers to go after the beaver, but fur prices are too low," says Ed Engen, of the Manitoba Department of Natural Resources. He says the beaver population has expanded because the antifur lobby has cut down the demand for fur. "The animal-rights people get emotional and say we're killing Canada's national symbol." He also blames warmer winters for a drop in demand for fur coats. Manitoba trappers caught only 10,000 beavers last year. With fur prices at $11 to $17 a pelt (compared with $68 or higher in the past decade), they may catch even fewer this year - this in a country built on the beaver fur trade. Beavers dam streams to enlarge the moat around their lodges. The province of Manitoba has spent $282,500 blowing up beaver dams and lodges. But a family of busy beavers can rebuild a 10-foot gash in a dam overnight. They take up lodgings in the side of streams until they rebuild their houses. And it is not just in sparsely populated Manitoba that Castor Canadensis is thriving. Beavers have adapted well to suburban living in the most heavily populated part of Canada, southwestern Ontario. Northwest of Toronto, beavers dam the little streams and creeks that flow into the Credit River. Just 20 miles due north of the main runway at Toronto's airport, beavers have flooded a stand of fir trees. "I've tried everything to get rid of them," says Bob Baker, a forester with the Credit Valley Conservation Association whose job is to look after the trees. "But they keep coming back and they've drowned a lot of trees." Beavers can weigh as much as 80 pounds. Their only natural enemies are black bears and wolves, and both of them are pretty rare, especially in Ontario. A big beaver could kill a dog or a coyote. With few natural enemies and their fur unwanted, the beaver is fat, happy, and multiplying, with four kits in a litter who are thrown out of the lodge after two years and go off to build their own homes. "Beavers are expanding their range and there are probably more beavers in this part of Ontario than there ever were," says Mr. Baker. Manitoba's battle against the beaver appears hopeless. A loser in the battle of the beaver, Baker has just about given up. But he looks on the bright side. "The beavers create a lot of new wetlands and that is good news for a lot of wildlife," says Baker, whose conservation authority is taking a more benign attitude to the industrious and sometimes destructive beavers. "If a landowner complains," he says, "we'll give him the name of a trapper. Otherwise we leave them alone."