8,479 research outputs found
Balanced binary trees in the Tamari lattice
We show that the set of balanced binary trees is closed by interval in the
Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are
balanced trees are isomorphic as posets to a hypercube. We introduce tree
patterns and synchronous grammars to get a functional equation of the
generating series enumerating balanced tree intervals
Pattern avoidance in binary trees
This paper considers the enumeration of trees avoiding a contiguous pattern.
We provide an algorithm for computing the generating function that counts
n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this
counting mechanism, we study the analogue of Wilf equivalence in which two tree
patterns are equivalent if the respective n-leaf trees that avoid them are
equinumerous. We investigate the equivalence classes combinatorially. Toward
establishing bijective proofs of tree pattern equivalence, we develop a general
method of restructuring trees that conjecturally succeeds to produce an
explicit bijection for each pair of equivalent tree patterns.Comment: 19 pages, many images; published versio
Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph
In this paper, we address the problem of enumerating all induced subtrees in
an input k-degenerate graph, where an induced subtree is an acyclic and
connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for
any its induced subgraph has a vertex whose degree is less than or equal to k,
and many real-world graphs have small degeneracies, or very close to small
degeneracies. Although, the studies are on subgraphs enumeration, such as
trees, paths, and matchings, but the problem addresses the subgraph
enumeration, such as enumeration of subgraphs that are trees. Their induced
subgraph versions have not been studied well. One of few example is for
chordless paths and cycles. Our motivation is to reduce the time complexity
close to O(1) for each solution. This type of optimal algorithms are proposed
many subgraph classes such as trees, and spanning trees. Induced subtrees are
fundamental object thus it should be studied deeply and there possibly exist
some efficient algorithms. Our algorithm utilizes nice properties of
k-degeneracy to state an effective amortized analysis. As a result, the time
complexity is reduced to O(k) time per induced subtree. The problem is solved
in constant time for each in planar graphs, as a corollary
Hopf structures on the multiplihedra
We investigate algebraic structures that can be placed on vertices of the
multiplihedra, a family of polytopes originating in the study of higher
categories and homotopy theory. Most compelling among these are two distinct
structures of a Hopf module over the Loday-Ronco Hopf algebra.Comment: 24 pages, 112 .eps file
Binary Decision Diagrams: from Tree Compaction to Sampling
Any Boolean function corresponds with a complete full binary decision tree.
This tree can in turn be represented in a maximally compact form as a direct
acyclic graph where common subtrees are factored and shared, keeping only one
copy of each unique subtree. This yields the celebrated and widely used
structure called reduced ordered binary decision diagram (ROBDD). We propose to
revisit the classical compaction process to give a new way of enumerating
ROBDDs of a given size without considering fully expanded trees and the
compaction step. Our method also provides an unranking procedure for the set of
ROBDDs. As a by-product we get a random uniform and exhaustive sampler for
ROBDDs for a given number of variables and size
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