Problems of finding distributively, the centers and medians of asynchronous communication networks are considered, and efficient distributed algorithms for determining these parameters are presented. The principal results of this paper are: 1. We show that for certain tree structures, the algorithms of Korach et.al.[7] fail to arrive at a correct solution, for both synchronous and asynchronous tree networks. Furthermore, their algorithms which are centralized, fail to terminate for such tree structures. We present modifications to their basic algorithm to ensure correctness and termination. 2. A new decentralized distributed algorithm is presented to determine these topological parameters. This is the first decentralized algorithm to find centers and medians reported in literature. The centers are determined in hmax(S) + D(T)/2 time using at most 3(n-1) + D(T)/2 messages where hmax(S) is the maximum height of a subtree rooted at any start node. The time and message complexities for median finding algorithms, are also linear in the number of nodes in the tree. 3. Finally, we present extensions of these algorithms for weighted trees.
