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A General Bijective Algorithm for Trees
William Y. C. Chen
Proceedings of the National Academy of Sciences of the United States of America
Vol. 87, No. 24 (Dec., 1990), pp. 9635-9639
Published by: National Academy of Sciences
Stable URL: http://www.jstor.org/stable/2356467
Page Count: 5
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Trees are combinatorial structures that arise naturally in diverse applications. They occur in branching decision structures, taxonomy, computer languages, combinatiorial 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 defined 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 Schroder trees. The main result gives a bijection between Schroder 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.
Proceedings of the National Academy of Sciences of the United States of America © 1990 National Academy of Sciences