The "vast woodlands" of New England in the early colonial period is not an accurate depiction of what the region was like, at least not in the sense that the settlers were surrounded by unbroken forest. I will leave it to an expert in early American folklore to talk about beliefs, but it's worth providing some additional context for what the physical and ecological settings of New England were during the colonial period.

The region, while it did host large forests, was at the time of European arrival a patchwork of woodlands of various ages, as well as agricultural land, all of which had been managed and shaped by a substantial native population. What is now New England has been recently estimated to have hosted up to 100,000 native inhabitants circa 1600. That may not sound like much, but the white population of New England would not hit that mark until 1700, a solid century later. The Native population of course would decline precipitously over this same period from epidemics (a "Great Dying" in 1616-1619 significantly reduced the coastal population) as well as warfare, displacement and enslavement at the hands of European settlers throughout the century.

Anyway, the native population significantly shaped the New England environment in which they lived. Coastal areas were cleared and cultivated - the area around Massachusetts Bay had less trees in 1630 than today. While native peoples relied on hunting (especially in what is now northern New England), native villages also grew crops, especially the "Three Sisters" of corn (maize), beans and squash.

The Pilgrims in Plymouth survived their first winter by taking corn caches and grave goods that they found in abandoned villages, and subsequent settlers often re-used native agricultural areas: pre-European settlement agricultural formations were visible even in Salem Massachusetts into the 20th century.

The New England forests as they existed had been managed by native peoples via controlled burning for centuries - small seasonal burns did not kill larger trees, but cleared out the underbrush and promoted new growth favored by game species such as deer and turkey. After arrival, the forests were treated by European colonists as a seemingly inexhaustible resource for profit and domestic use - ownership of land hinged in law on "improvement", which often meant at the minimum the felling of timber. Lumbering operations (commercial lumbering operations were set up in the mid 17th century in Maine and New Hampshire) often selectively chose tree species for particular purposes: white pines were prized for masts, pitch pines for pitch and naval stores, white oak for barrels (especially to be shipped to West Indian sugar plantations) and cedars and chestnut for outdoor wooden structures. The demand for firewood alone was insatiable - one estimate gives 260 million cords of wood burned between 1630 and 1800. Lumbering practices were often selective and wasteful, and even by the 18th century, writers noticed the disappearance of choice tree species across New England, with whole areas deforested. Timothy Dwight, traveling in the region in the 1790s, noted that in the 240 miles between Boston and New York City, a traveler would pass through no more than twenty miles of wooded land, in fifty or sixty parcels.

Sources:

Charles C. Mann, 1491, but especially

William Cronon, Changes in the Land: Indians, Colonists, and the Ecology of New England, which I cannot recommend enough on the subject of New England colonial ecology.