We provide an exact and approximation algorithm based on Dynamic Programming and an approximation algorithm based on Mixed Integer Programming for optimizing for the so-called dendritic connectivity on tree-structured networks in a multi-objective setting. Dendritic connectivity describes the degree of connectedness of a network. We consider different variants of dendritic connectivity to capture both network connectivity with respect to long and short-to-middle distances. Our work is motivated by a problem in computational sustainability concerning the evaluation of trade-offs in ecosystem services due to the proliferation of hydropower dams throughout the Amazon basin. In particular, we consider trade-offs between energy production and river connectivity. River fragmentation can dramatically affect fish migrations and other ecosystem services, such as navigation and transportation. In the context of river networks, different variants of dendritic connectivity are important to characterize the movements of different fish species and human populations. Our approaches are general and can be applied to optimizing for dendritic connectivity for a variety of multi-objective problems on tree-structured networks.