There's multiple conditions that contribute so that trees can not grow in the tundra, but the most important one is the climate.
The climate in the tundra is very inhospitable. Most of the year is winter, and the soil is frozen, not just on the surface, but also deep into the ground, thus not allowing a tree to be able to develop a root system. Also, the winds are very cold and strong, so a young tree will not be able to withstand them and will break. The summer lasts for only few weeks, and it is not a proper summer, but relatively cold one, thus not enabling the conditions for a tree to be able to vegetate.