We consider data gathering by a network with a sink node and a tree communication structure, where the goal is to minimize the total transmission cost of transporting the information, collected by the nodes, to the sink node. This problem requires a joint optimization of the data representation at the nodes and of the transmission structure. First, we study the case when the measured data are correlated random variables, both in the lossless scenario with Slepian-Wolf coding, and in the high-resolution lossy scenario with optimal rate-distortion allocation. We show that the optimal transmission structure is the shortest path tree, and we find, in closed-form, the rate and distortion allocation. Second, we study the case when the measured data are deterministic piecewise constant signals, and data is described with adaptive level wavelet-based multiresolution representation. We show experimentally that, when computation is decentralized, there is an optimal network division into node groups of adaptive size. Finally, we also analyze the node positioning problem where, given a correlation structure and an available number of sensors, the goal is to place the nodes optimally in terms of minimizing the transmission cost; our results show that important gains can be obtained compared to a uniformly distributed sensor positioning.