Suppose that each vertex of a graph $$G$$ G is either a supply vertex or a demand vertex and is assigned a supply or a demand. All demands and supplies are nonnegative constant numbers in a steady network, while they are functions of a variable $$\lambda $$ λ in a parametric network. Each demand vertex can receive “power” from exactly one supply vertex through edges in $$G$$ G . One thus wishes to partition $$G$$ G to connected components by deleting edges from $$G$$ G so that each component has exactly one supply vertex whose supply is at least the sum of demands in the component. The “partition problem” asks whether $$G$$ G has such a partition. If $$G$$ G has no such partition, one wishes to find the maximum number $$r^*$$ r ∗ , $$0\le r^* <1$$ 0 ≤ r ∗ < 1 , such that $$G$$ G has such a partition when every demand is reduced to $$r^*$$ r ∗ times the original demand. The “maximum supply rate problem” asks to compute $$r^*$$ r ∗ . In this paper, we deal with a network in which $$G$$ G is a tree, and first give a polynomial-time algorithm for the maximum supply rate problem for a steady tree network, and then give an algorithm for the partition problem on a parametric tree network, which takes pseudo-polynomial time if all the supplies and demands are piecewise linear functions of $$\lambda $$ λ .