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The paper offers tree-structured connectivity between the peer entities over an infrastructure network as an abstraction to embody the application-oriented processing functions on peer-to-peer information flows. Tree reconfigurations are triggered when failures or security violations occur in the connectivity path between the peer nodes (e.g., increase in path delays, DOS attack on a node). Reconfigurations are also initiated when the application-level properties of information flows change (e.g., lower data accuracy for non-important contents). For Information Assurance settings, a tree structure elegantly supports the desired properties of separation and non-inference between the low-level network entities and the application entities. A connectivity management algorithm reconfigures a tree satisfying the desired topological properties (e.g., low geographic spread, low peer-to-peer distances). The algorithm provides redundancy in the connectivity paths along the temporal dimension (on top of a rich connectivity in the physical infrastructure -- as in overlay networks). Application case studies of tree-structured connectivity with QoS and security goals are also described.