  We study the estimation of flows on trees, a structured generalization of
isotonic regression. A tree flow is defined recursively as a positive flow
value into a node that is partitioned into an outgoing flow to the children
nodes, with some amount of the flow possibly leaking outside. We study the
behavior of the least squares estimator for flows, and the associated minimax
lower bounds. We characterize the risk of the least squares estimator in two
regimes. In the first regime the diameter of the tree grows at most
logarithmically with the number of nodes. In the second regime, the tree
contains long paths. The results are compared with known risk bounds for
isotonic regression.
