Given a tree G = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the "rooted subtree problem", is to find a maximum weight subtree, with a specified root, from a given set of subtrees. The second problem, called "the subtree packing problem", is to find a maximum weight packing of node disjoint subtrees chosen from a given set...

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 ...

The Maximum Agreement Subtree problem is the following: given two rooted trees whose leaves are drawn from the same set of items (e.g., species), find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic. We consider the case which occurs frequently in practice, i.e., the case when the trees are binary, and give an O(n logn) time algor...

We consider the problem of finding the most profitable subtree of a given tree with at most K nodes which is known as the Cardinality Subtree of a Tree Problem (CSTP). We present a new extended formulation with O(nK) two-indexed variables and O(nK) constraints. Each variable contains information on node inclusion as well as the order by which the node is included in the subtree solution. We sho...

A subtree is picked at random from the collection of all subtrees of a complete graph. What is the probability the subtree spans? We find the surprising answer this question and a few closely related questions.

The overlap graphs of subtrees of a tree are equivalent to subtree filament graphs, the overlap graphs of subtrees of a star are cocomparability graphs, and the overlap graphs of subtrees of a caterpillar are interval filament graphs. In this paper, we show the equivalence of many more classes of subtree overlap and subtree filament graphs, and equate them to classes of complements of cochordal...

We address the problem of estimating the Wost-Case Execution Time (WCET) for loop-bounded programs. It is well known that for efficiency reasons, current techniques take little or no account of infeasible paths in the symbolic execution tree, thus limiting their accuracy. To consider all infeasible paths, ie. to be path-sensitive, would generally entail an impractical full path enumeration of t...

Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR) and rooted subtree prune and regraft (rSPR). For a pair of leaf-labelled binary trees with n leaves, the maximum number of such moves required to transform one into the other is n−Θ( √ n), extending a result of Ding, ...