For the climate of the mountains named the Alps, see climate) for a region above the tree-line. This climate is colder at high elevations, due to the lapse rate of air: air will tend to get colder as it rises, since it expands. The dry adiabatic lapse rate is 10°C per km of elevation or altitude. Therefore, moving up 100 meters on a mountain is roughly equivalent to moving 80 kilometers (45' (0.75°) of latitude) towards the pole  (http://www.unep-wcmc.org/mountains/mountain_watch/pdfs/mountainEnvironments.pdf). This relationship is only approximate, however, since local factors such as proximity to oceans can drastically modify the climate.
There have been several attempts at quantifying what constitutes an alpine climate.
Climatologist Wladimir Köppen demonstrated a relationship between the Arctic and Antarctic tree lines and the 10°C summer isotherm; i.e., places where the average temperature in the warmest calendar month of the year is below 10°C cannot support forests. See Köppen climate classification for more information.
However, Otto Nordenskiöld theorized that winter conditions also play a role: His formula is W = 9 - 0.1 C, with W denoting the average temperature in the warmest month and C the average of the coldest month, both in degrees Celsius (this would mean, for example, that if a particular location had an average temperature of -20°C in its coldest month, the warmest month would need to average 11°C or higher for trees to be able to survive there). Nordenskiöld's line tends to run to the north of Köppen's near the west coasts of the Northern Hemisphere continents, south of it in the interior sections, and at about the same latitude along the east coasts of both Asia and North America. In the Southern Hemisphere, all of Tierra del Fuego lies outside the polar region in Nordenskiöld's system, but part of the island (including Ushuaia, Argentina) is reckoned as being within the Antarctic under Köppen's.
In 1947, Holdridge improved on these schemes, by defining biotemperature: the mean annual temperature, where all temperatures below 0°C are treated as 0°C (because it makes no difference to plant life, being dormant). If the mean biotemperature is between 1.5°C and 3°C  (http://www.personal.dundee.ac.uk/~amjones/dundee/36biomes.htm), Holdridge quantifies the climate as alpine (or subpolar, of the low temperature is caused by latitude).