Optimal source location to feed a lossy distribution tree

Given a large but finite tree network designed to distribute a commodity to some or all of its nodes, the lossy model takes into account the attenuation of the flow through the branches of the network. The capacity of the source needed to feed the network depends on the node to which it is attached. In a finite network, there must be at least one node where the required source capacity is not higher than that at any other node. This short paper proves that there can be only one such node or, at most, two adjacent nodes with the same value for the required source capacity. Regarding the required source capacity as a function over the set of nodes, it is shown that this function can have only one local minimum (a node where the value is not higher than at any adjacent node), and no local maximum. Based on these properties, search algorithms are outlined to locate the node with the optimal source capacity.