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An antimagic labeling of a graph G is a bijection from the set of edges E(G) to {1, 2, . . . , |E(G)|} such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to the edges incident to u. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than K 2 is antimagic and the conjecture remains open even for trees. Here we prove that caterpillars are antimagic by means of an O(n log n) algorithm.

A k-antimagic labeling of a graph G is an injection from E(G) to {1, 2, . . . , |E(G)| + k} such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to edges incident to u. We call a graph k-antimagic when it has a k-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel [13] conjectured that every simple connected graph other than K 2 is antimagic, but the conjecture is still open even for trees. Here we study k-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use algorithmic aproaches, i.e., constructive approaches, to prove that any caterpillar of order n is ( (n − 1)/2 − 2)-antimagic. Furthermore, if C is a caterpillar with a spine of order s, we prove that when C has at least (3s + 1)/2 leaves or (s − 1)/2 consecutive vertices of degree at most 2 at one end of a longest path, then C is antimagic. As a consequence of a result by Wong and Zhu [22], we also prove that if p is a prime number, any caterpillar with a spine of order p, p − 1 or p − 2 is 1-antimagic. *

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The optimal transformation of one tree into another by means of elementary edit operations is an important algorithmic problem that has several interesting applications to computational biology. Here we introduce a constrained form of this problem in which a partial mapping of a set of nodes (the "seeds") in one tree to a corresponding set of nodes in the other tree is given, and present efficient algorithms for both ordered and unordered trees. Whereas ordered tree matching based on seeded nodes has applications in pattern matching of RNA structures, unordered tree matching based on seeded nodes has applications in co-speciation and phylogeny reconciliation. The latter involves the solution of the planar tanglegram layout problem, for which a polynomial-time algorithm is given here.

Many knowledge representation mechanisms are based on tree-like structures, thus symbolizing the fact that certain pieces of information are related in one sense or another. There exists a well-studied process of closure-based data mining in the itemset framework: we consider the extension of this process into trees. We focus mostly on the case where labels on the nodes are nonexistent or unreliable, and discuss algorithms for closurebased mining that only rely on the root of the tree and the link structure. We provide a notion of intersection that leads to a deeper understanding of the notion of support-based closure, in terms of an actual closure operator. We describe combinatorial characterizations and some properties of ordered trees, discuss their applicability to unordered trees, and rely on them to design efficient algorithms for mining frequent closed subtrees both in the ordered and the unordered settings. Empirical validations and comparisons with alternative algorithms are provided.