21 research outputs found
Counting Planar Tanglegrams
Tanglegrams are structures consisting of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the two trees. We say that a tanglegram is planar if it can be drawn in the plane without crossings. Using a blend of combinatorial and analytic techniques, we determine an asymptotic formula for the number of planar tanglegrams with n leaves on each side
Analogies between the crossing number and the tangle crossing number
Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with leaves decreases the tangle crossing number by at most , and this
is sharp. Additionally, if is the maximum tangle crossing number of
a tanglegram with leaves, we prove
. Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in time. This lower bound may be tight, even for
tanglegrams with tangle crossing number .Comment: 13 pages, 6 figure
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
A Satisfiability-based Approach for Embedding Generalized Tanglegrams on Level Graphs
A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in computational biology to compare evolutionary histories of species. In this paper we present a formulation of two related combinatorial embedding problems concerning tanglegrams in terms of CNF-formulas. The first problem is known as planar embedding and the second as crossing minimization problem. We show that our satisfiability formulation of these problems can handle a much more general case with more than two, not necessarily binary or complete, trees defined on arbitrary sets of leaves and allowed to vary their layouts
On trees, tanglegrams, and tangled chains
International audienceTanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared
The largest crossing number of tanglegrams
A tanglegram consists of two rooted binary trees with the same
number of leaves, and a perfect matching between the two leaf sets. In a
layout, the tanglegrams is drawn with the leaves on two parallel lines, the
trees on either side of the strip created by these lines are drawn as plane
trees, and the perfect matching is drawn in straight line segments inside the
strip. The tanglegram crossing number of is the
smallest number of crossings of pairs of matching edges, over all possible
layouts of . The size of the tanglegram is the number of matching
edges, say . An earlier paper showed that the maximum of the tanglegram
crossing number of size tanglegrams is ; but is
at least for infinitely many .
Now we make better bounds: the maximum crossing number of a size tanglegram
is at most , but for infinitely many ,
at least . The problem shows
analogy with the Unbalancing Lights Problem of Gale and Berlekamp
Tangled up in Tanglegrams
Tanglegrams are graphs consisting of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. A Tanglegram drawing is a special way of drawing a Tanglegram; and a Tanglegram is called planar if it has a drawing such that the matching edges do not cross. In this thesis, we will discuss various results related to the construction and planarity of Tanglegrams, as well as demonstrate how to construct all the Tanglegrams of size 4 by looking at two types of rooted binary trees - Caterpillar and Complete Binary Trees. After augmenting a Tanglegram with an edge between its roots, we will prove that the Tanglegram crossing number of the original Tanglegram is greater than or equal to the crossing number of the augmented Tanglegram taken as a graph. We will show that the removal of a matching edge from a Tanglegram of size n ≥3 decreases the Tanglegram crossing number by at most n − 3, and give a family of 1-edge panar Tanglegrams (one for every n ≥3) of size n with Tangle crossing number n − 3, showing that the previous statement is sharp. We will also discuss various conditions on the nonplanarity of Tanglegrams
On trees, tanglegrams, and tangled chains
Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared