To America with the Pilgrim Fathers came the dandelion, shepherd's purse, stinging nettle, couch grass, chickweed, groundsel and ubiquitous plantain. These and other weeds can be benevolent or toxic, says Mea Allen, an English gardener. Her garden contains many plants as old as three centuries, having belonged to Robert and Thomas Tradescant, kinsmen of the famous English botanist John Tradescant and his son John.

Allen is the author of a very good new book, "Weeds -- The Unbidden Guests in Our Gardens," published by The Viking Press, 191 pages, well illustrated, $14.95.

Some weeds, including poison ivy, are dangerous to man, she writes. Some like dodder are parasitic, living on other plants. Others excrete harmful substances which depress the growth of neighboring plants, and some become transit hotels for pests which overwinter in them.

Weeds can rouse the gardener's wrath and make him or her despair of ever winning the battle.But there are ways of eradicating them or at least controlling them, she says, although some ways are safe and some are dangerous.

On the other side, a study is revealing that some weeds may actually be worth encouraging for the help they give to garden plants, both by benefiting their growth and by quelling or killing the parts that attack them, and even by killing more pernicious weeds.

Weed seed are dispersed by birds, squirrels, all sorts of other animals and by the wind, Allen says. A bird flying to its wintering areas in Central and South America, a distance of 5,000 to 7, 000 miles, can take weed seeds along. The bobolink from North American meadowlands flies 5,000 miles to Bolivia, Paraguay or Brazil, the Arctic tern from the north of Greenland to the edge of the Antarctic pack-ice, 11,000 miles.

Alfred Newton, professor of zoology at Cambridge, sent Charles Darwin the leg of a partridge with a hard ball of earth weighing 6 1/2 ounces adhering to it. Darwin kept the earth for three years, but when he broke it up and placed it under a bell-glass no fewer than 82 plants grew from it.

In California, the man who probably was the greatest of all hybridizers, Luther Burbank, looked at three daisies: the on-eye daisy (weed) growing on the hills of Massachusetts, which was small, tenacious and hardy; an English Michaelmas daisy which was larger and coarser in stem; and the Japanese daisy, not large but of exquisite and almost dazzling whiteness.

He decided to marry the three and create a queen daisy that would have a slender stem, but firm, at least two feet tall and free from branches, with a flower larger than any daisy ever seen before, and petals of the purest white.

From 300,000 seeds Burbank grew 100,000 possible candidates and selected again. The seeding and selecting process was repeated for eight years.

At last, after growing millions of daisies, he achieved his queen, a flower of remarkable beauty with brilliant white petals of great size, the center pure yellow, the stems long and graceful.

He named it for Mount Shasta, which he had always revered, the snow-capped peak of the high Sierras which is one of the conspicuous landmarks of California.

In her book, Allen provides a unique identification guide covering 200 plants, all illustrated with clear line drawings, and gives her suggestions for control of the weeds.