Summary: Schrödinger operators on graphs with weighted edges may be defined using possibly infinite systems of ordinary differential operators. This work mainly considers radial trees, whose branching and edge lengths depend only on the distance from the root vertex. The analysis of operators with radial coefficients on radial trees is reduced, by a method analogous to separation of variables, to nonclassical boundary value problems on the line with interior point conditions. This reduction is used to study selfadjoint problems requiring boundary conditions ‘at infinity’.