An investigation was performed to establish the mathematical theories of a vibration for the plant growth and a wave distribution of a plant population on the boundary condition of a limiting factor in the environment. The mathematical theories of the plant growth vibration and wave distribution had been elucidated by the plant growth and the timber line on the middle slope of the west side of Mt. Paektu. The Betula ermaruii composes the timber line on about 2,060 m elevation of sea label, has a growth vibration on the ground surface and takes a wave distribution due to a boundary condition of alpine temperature gradient.