The Number of Plane Trees

Harary, Frank; Prins, Geert; Tutte, W. T.

doi:10.1016/s1385-7258(64)50038-4

Cited by 72 publications

(55 citation statements)

References 4 publications

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“…Although it could also follow from Corollary 3.3, it did not seem to be paid enough attention in the study of enumeration of plane trees. For example, from this point of view it would be obvious to obtain the number of unlabeled plane trees with a given degree type (6,7). COROLLARY 4.6 (ERDtLYI-ETHERINGTON).…”

Section: Plane Treesmentioning

confidence: 99%

A general bijective algorithm for trees.

Chen

1990

Proc. Natl. Acad. Sci. U.S.A.

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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.

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Section: Plane Treesmentioning

confidence: 99%

A general bijective algorithm for trees.

Chen

1990

Proc. Natl. Acad. Sci. U.S.A.

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“…These maps are the "rooted plane trees" of n edges [2]. If we write/o for the sum of the terms of ƒ not involving any of the y iy then T(n) is the coefficient of x 2n in/ 0 .…”

Section: Figure IVmentioning

confidence: 99%

On the enumeration of planar maps

Tutte¹

1968

Bull. Amer. Math. Soc.

151

107

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“…Consider any tree (see the Appendix of [8] for diagrams of the trees and identity trees with at most 12 points) and draw it in the plane. The result is called a plane tree: such trees were counted in [9]. Now draw a polygon joining [3] Graphs derived from trees 209 consecutive endpoints of the plane tree.…”

Section: Cubic Identity Graphsmentioning

confidence: 99%

Cubic identity graphs and planar graphs derived from trees

Balaban

Davies

Harary

et al. 1970

J. Aust. Math. Soc.

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The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

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The Number of Plane Trees

Cited by 72 publications

References 4 publications

A general bijective algorithm for trees.

A general bijective algorithm for trees.

On the enumeration of planar maps

Cubic identity graphs and planar graphs derived from trees

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