Despite only 1,3 % of Icelands total area is covered with trees, Iceland has some forest, and couple of them are natural. When the vikings came first to Iceland in the year 874, all of Icelandic lowlands were covered with forest, though only birch, rowan and willow, the forest coverage was around 25-40%. Through the centuries serious deforestration, along with heavy grazing (the sheep eats the young plant that are trying to grow), and countless volcanic eruptions as well as wind erosion has all helped to decrease the forest or tree coverage to almost nothing in the end of 19th century. Since then, natural birch forests that survived have been let alone from grazing, and many millions of trees: spruces, pine, larch, birch, willow, ash, elm, maples, aspens etc. from many places around the world have been planted with great success. These small forests are scattered all around the island, but the biggest is Hallormsstaðaskógur, at the lake of Lögurinn (30 km northwest of Egilsstaðir) in Eastern part of Iceland, around 700 km from Reykjavik). The natural birch forest (the little who remained) was saved in 1905 when the forest was protected by law from the Icelandic parliament. Many foreign trees have been planted, though notably larch and pine, as well as spruce of several species. Larch is well suited to the climate there, which is known for its good mild summers (temp. 10-20 C°, on good day it can reach 25 C°) and Fljótdalshérað region (where this forest is), has continential climate character, with good summers but cold winters. This forest is the largest in Iceland, 15 km long, and around 2000 hectares. It has beautiful surroundings, it is sheltered by mountains and Föhn winds makes this place warm. If you are travelling around Iceland and get little bit annoyed about the lack of trees, you can always stop and camp in one of the countries forests or woodlands, like Hallormsstaðaskogur. To see a map of open forests for visitors: http://www.skogur.is/Pages/60
Here are some pictures from Hallormsstadaskógur.