Given a tree with nonnegative edge cost and nonnegative
vertex weight, and a number $k\geq0$, we consider the following four cut problems: cutting vertices of weight at most or at least $k$ from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The {\rm MinMstCut} problem (cut vertices of weight \emph{at most} $k$ and \emph{minimize} the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard.
In this paper, we design an $O(nl/\varepsilon)$-time
$O(l^2/\varepsilon+n)$-space algorithm for {\rm MaxMstCut},
and $O(nl(1/\varepsilon+\log n))$-time
$O(l^2/\varepsilon+n)$-space algorithms for the other two problems,
{MinLstCut} and {\rm MaxLstCut},
where $n$ is the number of vertices in the tree, $l$ the number of leaves,
and $\varepsilon>0$ the prescribed error bound.