In a distributed stream processing system, streaming data are continuously disseminated from the sources to the distributed processing servers. To enhance the dissemination efficiency, these servers are typically organized into one or more dissemination trees. In this paper, we focus on the problem of constructing dissemination trees to minimize the average loss of fidelity of the system. We observe that existing heuristic-based approaches can only explore a limited solution space and hence may lead to sub-optimal solutions. On the contrary, we propose an adaptive and cost-based approach. Our cost model takes into account both the processing cost and the communication cost. Furthermore, as a distributed stream processing system is vulnerable to inaccurate statistics, runtime fluctuations of data characteristics, server workloads, and network conditions, we have designed our scheme to be adaptive to these situations: an operational dissemination tree may be incrementally transformed to a more cost-effective one. Our adaptive strategy employs distributed decisions made by the distributed servers independently based on localized statistics collected by each server at runtime. For a relatively static environment, we also propose two static tree construction algorithms relying on apriori system statistics. These static trees can also be used as initial trees in a dynamic environment. We apply our schemes to both single- and multi-object dissemination. Our extensive performance study shows that the adaptive mechanisms are effective in a dynamic context and the proposed static tree construction algorithms perform close to optimal in a static environment.