  We consider problems in which we are given a rooted tree as input, and must
find a subtree with the same root, optimizing some objective function of the
nodes in the subtree. When this function is the sum of constant node weights,
the problem is trivially solved in linear time. When the objective is the sum
of weights that are linear functions of a parameter, we show how to list all
optima for all possible parameter values in O(n log n) time; this parametric
optimization problem can be used to solve many bicriterion optimizations
problems, in which each node has two values xi and yi associated with it, and
the objective function is a bivariate function f(SUM(xi),SUM(yi)) of the sums
of these two values. A special case, when f is the ratio of the two sums, is
the Weighted Maximum-Mean Subtree Problem, or equivalently the Fractional
Prize-Collecting Steiner Tree Problem on Trees; for this special case, we
provide a linear time algorithm for this problem when all weights are positive,
improving a previous O(n log n) solution, and prove that the problem is
NP-complete when negative weights are allowed.
