The performance of district heating networks (DHNs) to provide heat to users under different operation conditions depends largely on their topology. It is of great significance to analyze the topological and functional characteristics of DHNs and propose design and reconstruction guidelines for vulnerability reduction. This study proposes binary tree and generalized tree models of DHNs, from which their topological characteristics can be quickly identified, and the functional analysis can be facilitated. Then, we propose a topology design and reconstruction framework for DHNs based on vulnerability assessment. Various characteristics of the branched and looped structure are studied in their influence on system vulnerability, based on which several design criteria for single-source DHNs are deduced. Specifically, single-source branched DHNs shall be designed as symmetrical to avoid long pipelines bearing enormous heat loads. When transforming the branched DHNs into looped DHNs, priority should be given to adding cotrees between the intermediate nodes of high importance, and the diameters of looped pipes should be enlarged. Finally, a detailed case study on a real-world single-source DHN demonstrates the effectiveness of the proposed approaches and design criteria.