Original Research Article

Article volume = 2024 and issue = 2

Pages: 152–168

Article publication Date: August 18, 2024

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# Enumeration of k-plane trees and forests

#### Albert Oloo Nyariaro(a) and Isaac Owino Okoth(b)*

(a) Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya.

(b) Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya.

##### Abstract:

A k-plane tree is an ordered tree in which the vertices are labelled by integers {1, 2, . . . , k} and satisfies the condition i + j ⩽ k + 1 where i and j are adjacent vertices in the tree. These trees are known to be counted by Fuss-Catalan numbers. In this paper, we use generating functions and decomposition of trees to enumerate these trees according to degree of the root, label of the first child of the root and number of forests of k-plane trees. The results of this paper generalize known results for 2-plane trees and plane trees.

##### Keywords:

k-plane tree, degree, first child, forest.

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##### Cite this article as:

- Albert Oloo Nyariaro and Isaac Owino Okoth*, Enumeration of k-plane trees and forests, Communications in Combinatorics, Cryptography & Computer Science, 2024(2), PP.152–168, 2024
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