In most cooperative communication works, perfect synchronization among relay nodes is assumed in order to achieve the cooperative diversity. However, this assumption is not realistic due to the distributed nature of each relay node. In this paper, we propose a family of distributed space-time trellis code (DSTTC) that does not require the synchronization assumption. It is shown that the proposed DSTTC has minimum memory order, and the construction of such DSTTCs is equivalent to using the generator matrices that have full row rank, regardless of sub-matrices shifting problem. Here, a sub-matrix corresponds to the generator matrix of a relay node under frequency-selective channels. We derive sufficient conditions on the code design such that the full cooperative and multipath diversities can be achieved. By further studying the diversity product, we design the DSTTCs which can achieve full diversity and the maximum coding gain through the exhaustive computer search. The newly proposed codes exhibit good properties, e.g., high energy efficiency and low synchronization cost, and can be applied to distributed wireless networks. Finally, various numerical examples are provided to corroborate the analytical studies.