  The Schr\"odinger operator on a metric tree is a family of ordinary
differential operators on its edges complemented by certain matching conditions
at the vertices. The regular trees are highly symmetric. This allows one to
construct an orthogonal decomposition of the space L_2 on the tree which
reduces the Schr\"odinger operator with any symmetric weight. Using this
decomposition, we analyse the spectrum of such operators, including the free
Laplacian, under various assumptions about the tree and the potential.
