The Short Answer: I’m answering this question a tad late, in mid-April, so most of the snow in the U.S. is gone, but as deep snow melts, it often melts first around the base of trees, as Raymond mentions. Notice that a snow ring shows in this photo of a flower pot. This is a clue that it’s probably not that the trees are generating heat biologically, although some plants are capable of doing that. What happens with trees is that the tree branches keep some snow from reaching the ground. Suspended in the branches, that snow is subjected to more sun than snow distributed on the ground so it melts and never falls all the way to the ground under the tree. Also, when the sun is shining, the tree absorbs more heat than the surrounding snow. This happens because the tree is darkly colored — especially after the snow has melted so that the dark branches or evergreen needles are exposed. The tree trunk gets warm, and then radiates some of that heat back out, which melts the snow around the base. The same thing happens with the flower pot and other objects like telephone poles and fence posts.
More Information: This phenomenon occurs under deciduous trees that have dropped their leaves, and even more under evergreens because they catch more snow, and the dark color of their needles helps them absorb more heat, compared to deciduous trees, which are generally lighter colored. In fact, in northern evergreen forests that get large amounts of snow, the “snow wells” around trees can be two meters (6 feet) deep or more. This represents a serious danger to hikers and skiers who sometimes fall into the wells and can’t get back out.
This website gives information on the dangers of snow wells and tips for escape: http://www.deepsnowsafety.org/index.php/
Snow well paradox: Scientists study snow wells because the phenomenon that creates them has a significant effect on how heat and water are absorbed in northern forests. In their studies, they’ve found something surprising. Though the warmth radiating from the tree often melts the snow around the trunk, that same area also experiences the deepest frost. How does that happen? The bare ground at the base of the tree is exposed to the frigid winter air, especially at night or in cloudy weather when the sun is not warming the tree. The surrounding areas, covered by snow that is often quite deep, are insulated from the worst of the cold. Though the snow melts around the tree, paradoxically, that’s also where the frost goes deepest.