154 research outputs found
Analysis on the reconstruction accuracy of the Fitch method for inferring ancestral states
Fitch method for inferring ancestral state
Inducibility of d-ary trees
CITATION: Czabarka, E. et al. 2020. Inducibility of d-ary trees. Discrete Mathematics, 343(2). doi:10.1016/j.disc.2019.111671.The original publication is available at https://www.sciencedirect.com/journal/discrete-mathematicsImitating the binary inducibility, a recently introduced invariant of binary trees (Cz-
abarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d â„ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum
density of a d-ary tree D in T and the inducibility of D is of order O(|T |â1/2) compared
to the general case where it is shown that the difference is O(|T |â1) which, in particular,
responds positively to a conjecture on the inducibility in binary trees. We also discover
that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we
establish a general lower bound on the inducibility and also provide a bound for some
special trees. Moreover, we find that the maximum inducibility is attained for binary
caterpillars for every d.https://www.sciencedirect.com/science/article/pii/S0012365X19303498Publishers versio
Rainbow Turan Methods for Trees
The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur Ìan number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work
A 'stochastic safety radius' for distance-based tree reconstruction
A variety of algorithms have been proposed for reconstructing trees that show
the evolutionary relationships between species by comparing differences in
genetic data across present-day taxa. If the leaf-to-leaf distances in a tree
can be accurately estimated, then it is possible to reconstruct this tree from
these estimated distances, using polynomial-time methods such as the popular
`Neighbor-Joining' algorithm. There is a precise combinatorial condition under
which distance-based methods are guaranteed to return a correct tree (in full
or in part) based on the requirement that the input distances all lie within
some `safety radius' of the true distances. Here, we explore a stochastic
analogue of this condition, and mathematically establish upper and lower bounds
on this `stochastic safety radius' for distance-based tree reconstruction
methods. Using simulations, we show how this notion provides a new way to
compare the performance of distance-based tree reconstruction methods. This may
help explain why Neighbor-Joining performs so well, as its stochastic safety
radius appears close to optimal (while its more classical safety radius is the
same as many other less accurate methods).Comment: 18 pages, 1 figure, 4 table
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size in the binary search
tree and the random recursive tree (of total size ) asymptotically has a
Poisson distribution if , and that the distribution is
asymptotically normal for . Furthermore, we prove similar
results for the number of subtrees of size with some required property , for example the number of copies of a certain fixed subtree . Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of -protected nodes in a binary search tree or random recursive tree.
The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings
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