  In this paper we consider the linear Schrodinger equation (LSE) on a regular
tree with the last generation of edges of infinite length and analyze some
unique continuation properties. The first part of the paper deals with the LSE
on the real line with a piece-wise constant coefficient and uses this result in
the context of regular trees. The second part treats the case of a LSE with a
real potential in the framework of a star-shaped graph.
