Bisection of trees and sequences

Bisection of trees and sequences

Alon, N., Y. Caro and I. Krasikov, Bisection of trees and sequences, Discrete Mathematics 114 (1993) 337. A graph G is called bisectable if it is an edge-disjoint union of two isomorphic subgraphs. We show that any tree T with e edges contains a bisectable subgraph with at least e O(e/log log e) edges. We also show that every forest of size e, each component of which is a star, contains a bisec...

A bisection algorithm for grammar-based compression of ordered trees

Simulating sequences and phylogenetic trees

Using the seqgen() function available in the phyclust package [1], data were simulated by mutating DNA sequences along phylogenetic trees. The topology of the trees were generated at two stages via the ms program [1, 2]. At the first stage we randomly generated a root sequence and a star-like ancestor tree having equal branch lengths and tips equal to the number of desired clusters. The tip lab...

Trees and Meta-Fibonacci Sequences

For k > 1 and nonnegative integer parameters ap, bp, p = 1..k, we analyze the solutions to the meta-Fibonacci recursion C(n) = ∑k p=1 C(n − ap − C(n − bp)), where the parameters ap, bp, p = 1..k satisfy a specific constraint. For k = 2 we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include ...

Sequences Characterizing k-Trees

A non-decreasing sequence of n integers is the degree sequence of a 1-tree (i.e., an ordinary tree) on n vertices if and only if there are least two 1’s in the sequence, and the sum of the elements is 2(n − 1). We generalize this result in the following ways. First, a natural generalization of this statement is a necessary condition for k-trees, and we show that it is not sufficient for any k >...