We study network loss tomography based on observing average loss rates over a set of paths forming a tree – a severely underdetermined linear problem for the unknown link loss probabilities. We examine in detail the role of sparsity as a regularising principle, pointing out that the problem is technically distinct from others in the compressed sensing literature. While sparsity has been applied in the context of tomography, key questions regarding uniqueness and recovery remain unanswered. Our work exploits the tree structure of path measurements to derive sufficient conditions for sparse solutions to be unique and the condition that ℓ1 minimization recovers the true underlying solution. We present a fast single-pass linear algorithm for ℓ1 minimization and prove that a minimum ℓ1 solution is both unique and sparsest for tree topologies. By considering the placement of lossy links within trees, we show that sparse solutions remain unique more often than is commonly supposed. We prove similar results for a noisy version of the problem.