Visualising Populations of Rooted Labeled Trees ona
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A Combinatorial Proof of Postnikov's Identity and a Generalized Enumeration of Labeled Trees
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
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In this paper, we construct explicitly a noncommutative symmetric (NCS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the NCS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a ...
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Most supertree algorithms combine collections of rooted phylogenetic trees with overlapping leaf sets into a single rooted phylogenetic tree. It is implicit in all these algorithms that the leaves of the rooted phylogenetic trees in the input collection, as a whole, represent non-nested taxa. Thus, for example, the “domestic dog” and “mammal” cannot be represented by two distinct leaves in such...
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A chief problem in phylogenetics and database theory is the computation of a maximum consistent tree from a set of rooted or unrooted trees. A standard input are triplets, rooted binary trees on three leaves, or quartets, unrooted binary trees on four leaves. We give exact algorithms constructing rooted and unrooted maximum consistent supertrees in time O(2nm log m) for a set of m triplets (qua...
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We present an approach that allows performing computations related to the Baker-Campbell-Haussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. We develop a rewriting algorithm (based on labeled rooted tree...
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