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A discrete memoryless network (DMN) is a memoryless multiterminal channel with discrete inputs and outputs. A sequence of inner bounds to the DMN capacity region is derived by using code trees. Capacity expressions are given for three classes of DMNs: (1) a single-letter expression for a class with a common output, (2) a two-letter expression for a binary-symmetric broadcast channel (BC) with partial feedback, and (3) a finite-letter expression for push-to-talk DMNs. The first result is a consequence of a new capacity outer bound for common output DMNs. The third result demonstrates that the common practice of using a time-sharing random variable does not include all time-sharing possibilities, namely, time sharing of channels. Several techniques for improving the bounds are developed: (1) causally conditioned entropy and directed information simplify the inner bounds, (2) code trellises serve as simple code trees, (3) superposition coding and binning with code trees improves rates. Numerical computations show that the last technique enlarges the best known rate regions for a multiple-access channel (MAC) and a BC, both with feedback. In addition to the rate bounds, a sequence of inner bounds to the DMN reliability function is derived. A numerical example for a two-way channel illustrates the behavior of the error exponents.