Note on islands in path-length sequences of binary trees

An earlier characterization of topologically ordered (lexicographic) path-length sequences of binary trees is reformulated in terms of an integrality condition on a scaled Kraft sum of certain subsequences (full segments, or islands). The scaled Kraft sum is seen to count the set of ancestors at a certain level of a set of topologically consecutive leaves is a binary tree.Comment: 4 page

The permutation-path coloring problem on trees

AbstractIn this paper we first show that the permutation-path coloring problem is NP-hard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations, which are permutations that contain exactly one cycle, on trees with maximum degree greater than or equal to 4. We calculate a lower bound on the average complexity of the permutation-path coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutation-path coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n/4+o(n). We extend these results and obtain an upper bound on the average complexity of the permutation-path coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutions-path coloring problem on arbitrary trees

Fringe trees, Crump-Mode-Jagers branching processes and mm-ary search trees

This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees. We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of $m$-ary search trees in detail; this seems to be new. Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for $m$-ary search trees, and give for example new results on protected nodes in $m$-ary search trees. A separate section surveys results on height, saturation level, typical depth and total path length, due to Devroye (1986), Biggins (1995, 1997) and others. This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees

ローテーションの回数に基づく2分木間の距離

A certain distance between two binary trees is defined. The distance is the minimal number of the rotations changing from tree A to tree B．If the distance between tree A and tree B is 1, we say that the two trees are connected. Binary trees are expressed by the codewords as the computer representations, and ranked by the lexicographic generation of codewords. An algorithm that makes the table of the connection using the rank is proposed. The distance is expressed by the minimal path length on the graph made from the table

Distorted Metrics on Trees and Phylogenetic Forests

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances {d(u, v)} between leaves. It is well known that these determine the tree and the d length of all edges. Here we consider distortions ˆd of d such that for all leaves u and v it holds that |d(u, v)− ˆd(u, v)| \u3c f/2 if either d(u, v) \u3c M or ˆd(u, v) \u3c M, where d satisfies f ≤ d(e) ≤ g for all edges e. Given such distortions we show how to reconstruct in polynomial time a forest T1, . . . , Tα such that the true tree T may be obtained from that forest by adding α − 1 edges and α − 1 ≤ 2−Ω(M/g)n. Metric distortions arise naturally in phylogeny, where d(u, v) is defined by the log-det of a covariance matrix associated with u and v. When u and v are “far”, the entries of the covariance matrix are small and therefore dˆ(u, v), which is defined by log-det of an associated empirical correlation matrix may be a bad estimate of d(u, v) even if the correlation matrix is “close” to the covariance matrix. Our metric results are used in order to show how to reconstruct phylogenetic forests with small number of trees from sequences of length logarithmic in the size of the tree. Our method also yields an independent proof that phylogenetic trees can be reconstructed in polynomial time from sequences of polynomial length under the standard assumptions in phylogeny. Both the metric result and its applications to phylogeny are almost tight