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Knowing how important a node or an edge is, within a network, can be very valuable. In the area of complex networks, a variety of centrality measures, which assign to each node or edge a number representing its importance, have been proposed. In this paper, we consider two such measures, namely, node and edge betweenness that characterize how often a node or edge lies on the shortest paths between all pairs of nodes. For each measure, we use a dynamical systems approach to develop continuous- and discrete-time distributed algorithms, which enable every node in an undirected and unweighted tree graph to compute its own measure with only local interaction and without any centralized coordination. We show that the algorithms are simple and scalable, with the continuous-time version being unconditionally exponentially convergent, and the discrete-time version unconditionally exhibiting a deadbeat response. Moreover, we show that they require minimal node memories to execute, bypass entirely the need to construct shortest paths, and can handle time-varying topologies.