A general bijective algorithm for trees.

Abstract

Trees are combinatorial structures that arise naturally in diverse applications. They occur in branching decision structures, taxonomy, computer languages, combinatorial optimization, parsing of sentences, and cluster expansions of statistical mechanics. Intuitively, a tree is a collection of branches connected at nodes. Formally, it can be defmed as a connected graph without cycles. Schroder trees, introduced in this paper, are a class of trees for which the set of subtrees at any vertex is endowed with the structure of ordered partitions. An ordered partition is a partition of a set in which the blocks are linearly ordered. Labeled rooted trees and labeled planed trees are both special classes of Schrdder trees. The main result gives a biection between Schrider trees and forests of small trees-namely, rooted trees of height one. Using this bijection, it is easy to encode a Schroder tree by a sequence of integers. Several classical algorithms for trees, including a combinatorial proof of the Lagrange inversion formula, are immediate consequences of this bijection.

Hook Length Polynomials for Plane Forests of a Certain Type

“…It's not hard to see that G r is connected. (1) and S (2) in V r , C F r, k, S (1) = C F r, k, S (2) .…”

“…There is a well-known result on the cardinality of F (r) [1,4,6,10], denoting by n = ∑ d≥1 r d = |I(F)| the number of internal vertices and = − ∑ d≥0 (d − 1)r d the number of trees in F: …”

“…We call two partitions S (1) and S (2) adjacent if there exists i ∈ [n − 1], such that we can obtain S (1) by swapping i and i + 1 in S (2) .…”

“…For any colored labelled forest F ∈ C F r, k, S (1) , let v 1 be the vertex with label i and v 2 be the vertex with label i + 1:…”

A parallel algorithm for generation of RNA secondary structures with length n and k base-pairs

“…It is proven that the total number of such trees can be counted using the Narayana numbers [6,7,19]:…”