  A symmetric relation in the probabilistic Green's function for birth-death
chains is explored. Two proofs are given, each of which makes use of the known
symmetry of the Green's functions in other contexts. The first uses as primary
tool the local time of Brownian motion, while the second uses the reciprocity
principle from electric network theory. We also show that the the second proof
extends easily to cover birth-death chains (a.k.a. state-dependent random
walks) on trees.
