  We address the problem,`Is a local tree structure sufficient for the local
optimality of message passing algorithm in low density parity check codes?'.It
is shown that the answer is negative. Using this observation, we pinpoint a
flaw in the proof of Theorem 1 in the paper `The Capacity of Low-Density
Parity-Check Codes Under Message-Passing Decoding' by Thomas J. Richardson and
R\"udiger L.Urbanke\cite{RUCapacity}. We further provide a new proof of that
theorem based on a different argument.
