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The number of plane trees with a given partition

  • F. Harary (a1) and W. T. Tutte (a2)

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This note is a continuation of the articles [6] and [2]. In [1], trees with a given partition α = (a1; a2, …), where ai is the number of vertices (points) of valency (degree) i were enumerated. After the determination of the number of plane trees in [2], the number of planted plane trees with a given partition α was found explicitly in [6]. In the present note, the number of plane trees with a given partition is expressed as a function of the number of planted trees with a given partition. The method, which is not new, consists of an application of the enumeration techniques of Otter [3] and Pólya [4]; it was used in [1] and also by Riordan [5].

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1. Harary, F. and Prins, G., “The number of homeomorphically irreducible trees, and other species”, Acta Math., 101 (1959), 141162.
2. Harary, F. and Prins, G., and Tutte, W. T., “The number of plane trees”, Nederl. Wetensch. Proc. Ser. A,67 = Indag, Math., 26 (1964), 319329.
3. Otter, R., “The number of trees”, Annals of Math., 49 (1948), 583599.
4. Pólya, G., “Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen”, Acta Math., 68 (1937), 145254.
5. Riordan, J.. “The numbers of labeled colored and chromatic trees”, Acta Math., 97 (1957), 211225.
6. Tutte, W. T., “The number of planted plane trees with a given partition”, American Math. Monthly, 71 (1964), 272277.
7. Read, R. C., “A note on the number of functional digraphs”, Math. Annalen, 143 (1961), 109110.
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The number of plane trees with a given partition

  • F. Harary (a1) and W. T. Tutte (a2)

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