A multi-hop relay network with multiple antenna terminals in a quasi-static slow fading environment is considered. The fundamental diversity-multiplexing gain tradeoff (DMT) is analyzed in the case of half-duplex relay terminals. While decode-and-forward (DF) relaying achieves the optimal DMT in the full-duplex relay scenario, it is shown that the dynamic decode-and-forward (DDF) protocol achieves the optimal DMT if the relay is constrained to half-duplex operation. For the latter case, static protocols are considered as well, and the corresponding DMT performances are shown to fall short of the optimal performance, which indicates that dynamic channel allocation is required for optimal DMT performance over half-duplex relay networks. The optimal DMT is expressed as the solution of a convex optimization problem and explicit DMT expressions are presented for some special cases.