We address the problem of localizing the source of infection in an undirected, tree-structured network under a susceptible-infected outbreak model. The infection propagates with independent random time increments (i.e., edge-delays) between neighboring nodes, while only the infection times of a subset of nodes can be observed. We show that a reduced set of observers may be sufficient, in the statistical sense, to localize the source and characterize its identifiability via the joint Laplace transform of the observers' infection times. Using the explicit form of these transforms in terms of the edge-delay probability distributions, we propose scale-invariant least-squares estimators of the source. We evaluate their performance on synthetic trees and on a river network, demonstrating accurate localization under diverse edge-delay models. To conclude, we highlight overlooked technical challenges for observer-based source localization on networks with cycles, where standard spanning-tree reductions may be ill-posed.