Birch trees are not portrayed in the How to Train Your Dragon Franchise. In reality, they are species belonging to the Genus Betula. They are characterized by whitish papery bark that often peels away from the trunk. They are deciduous, meaning they loose their leaves in the Winter. Birch grow on temperate and cool climates in the Northern Hemisphere on all three continents. They are also sometimes cultivated in areas with a temperate climate away from their natural distribution
In the Franchise, birch is mentioned as a tree involved in wedding ceremonies, called the "Betrothal Birch". Presumably the couple must chop down the Bethrothal Birch, though the significance is unknown.
In reality, birch trees have a wide range of uses. These include food, herbal medicine, wood, even makeshift paper.
Birch trees are mentioned by Spitelout Jorgenson in the episode "Snotlout Gets the Axe". Snotlout is to safely deliver the Jorgenson Ceremonial Axe to the Island of Friga for a marriage ceremony between a Jorgenson and a Hofferson, so they can chop down the "Betrothal Birch".