For a class of stationary Gaussian sources and the squared-error distortion measure, an asymptotically optimal tree coding scheme is derived using tree codes with finite branch length. The distribution of the reproduction process is derived in an explicit form, and using the random coding argument and results from the theory of branching processes with stationary ergodic environmental processes, the source coding theorem is proved. The theorem is applicable to all Gaussian sources with a power spectrum that satisfies the Lip 1 condition with no restriction on the coding rate.