  Reflection and transmission of waves in piecewise constant layered media are
important in various imaging modalities and have been studied extensively.
Despite this, no exact time domain formulas for the Green's functions have been
established. Indeed, there is an underlying combinatorial obstacle: the
analysis of scattering sequences. In the present paper we exploit a
representation of scattering sequences in terms of trees to solve completely
the inherent combinatorial problem, and thereby derive new, explicit formulas
for the reflection and transmission Green's functions.
