When is compress-and-forward optimal?

In many known examples where compress-and-forward (CF) for relay networks is capacity achieving, it is only trivially so, i.e., it falls back to hashing without quantization. A potentially better strategy is to decode as much as possible and to compress the residual information, i.e., a combination of decode-and-forward (DF) and CF (Cover and El Gamal´s Theorem 7). Indeed such a strategy was shown to be optimal by Kang and Ulukus for a certain class of diamond relay networks consisting of a source, a noisy relay, a noiseless relay, and a destination. In this paper, we discuss why it can be optimal for such channels. Furthermore, we generalize the result to a certain class of tree networks with an arbitrary number of nodes consisting of multiple cascaded diamond relay networks. We show that a combination of DF and CF is optimal for the network and its capacity is given by a simple expression. As in the diamond channel, the capacity is strictly less than the cut-set bound.