https://repository.maseno.ac.ke/handle/123456789/1935 2026-05-15T12:08:31Z 2026-05-15T12:08:31Z Kariuki, Yvonne Wakuthii Okoth, Isaac Owino Nyamwala, Fredrick Oluoch https://repository.maseno.ac.ke/handle/123456789/6225 2024-11-12T16:00:56Z 2024-07-02T00:00:00Z

Counting formulas and bijections of nondecreasing 2-noncrossing trees Kariuki, Yvonne Wakuthii; Okoth, Isaac Owino; Nyamwala, Fredrick Oluoch In this paper, we introduce nondecreasing 2-noncrossing trees and enumerate them according to their number of vertices, root degree, and number of forests. We also introduce nondecreasing 2-noncrossing increasing trees and count them by considering their number of vertices, label of the root, label of the leftmost child of the root, root degree, and forests. We observe that the formulas enumerating the newly introduced trees are generalizations of little and large Schroder ¨ numbers. Furthermore, we establish bijections between the sets of nondecreasing 2-noncrossing trees, locally oriented noncrossing trees, labelled complete ternary trees, and 3-Schroder paths.

2024-07-02T00:00:00Z Nyariaro, Albert Oloo Okoth, Isaac Owino https://repository.maseno.ac.ke/handle/123456789/6224 2024-11-12T15:55:03Z 2024-08-18T00:00:00Z

Enumeration of k-plane trees and forests Nyariaro, Albert Oloo; Okoth, Isaac Owino 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.

2024-08-18T00:00:00Z Kariuki, Yvonne Wakuthii Okoth, Isaac Owino Nyamwala, Fredrick Oluoch https://repository.maseno.ac.ke/handle/123456789/6146 2024-08-07T13:07:58Z 2024-08-08T00:00:00Z

On non-decreasing 2-plane trees Kariuki, Yvonne Wakuthii; Okoth, Isaac Owino; Nyamwala, Fredrick Oluoch In this paper, we have introduced the set of non-decreasing 2-plane trees. These are plane trees whose vertices receive labels from the set {1, 2} such that the sum of labels of adjacent vertices is at most 3 and that the labels of siblings are weakly increasing from left to right. We have obtained the formula for the number of these trees with a given number of vertices and label of the root. Further, we have obtained the number of these trees given root degrees and label of the eldest child of the root. We have also constructed bijections between the set of non-decreasing 2-plane trees with roots labelled 2 and the sets of little Schröder paths, plane trees in which leaves receive two labels, restricted lattice paths and increasing tableaux. For non-decreasing 2-plane trees with roots labelled 1, we have obtained bijections between the set of these trees and the sets of large Schröder paths and row-increasing tableaux.

2024-08-08T00:00:00Z Nyariaro, Albert Oloo Okoth, Isaac Owino https://repository.maseno.ac.ke/handle/123456789/6145 2024-08-07T12:58:38Z 2024-09-01T00:00:00Z

Bijections for classes of labelled trees. Nyariaro, Albert Oloo; Okoth, Isaac Owino Trees are acyclic connected graphs. Plane trees, d-ary trees, binary trees, noncrossing trees and their generalizations, which are families of trees, have been enumerated by many authors using various statistics. These trees are known to be enumerated by Catalan or Catalan-like formulas (Fuss-Catalan numbers). One of the most common approaches to the enumeration of these trees is by means of generating functions. Another method that can be used to enumerate them is by constructing bijections between sets of the same cardinality. The bijective method is preferred to other methods by many combinatorialists. So, in this paper, we construct bijections relating k-plane trees, k-noncrossing increasing trees, k-noncrossing trees, k-binary trees and weakly labelled k-trees.

2024-09-01T00:00:00Z Kariuki, Yvonne Wakuthii https://repository.maseno.ac.ke/handle/123456789/6061 2024-03-25T13:11:13Z 2023-10-19T00:00:00Z

Bijections of plane Husimi graphs and certain combinatorial structures Kariuki, Yvonne Wakuthii Plane Husimi graphs are combinatorial structures obtained when we replace edges in plane trees with complete graphs such that the resultant structures are connected and cycle free. The formula that counts these structures is known to enumerate other combinatorial structures. In this paper, we construct bijections between the set of plane Husimi graphs and the sets of plane trees, dissections of convex polygons, sequences satisfying certain properties, standard Young tableaux, Deutsch paths and restricted lattice paths. http://ejma.euap.org

2023-10-19T00:00:00Z Nyariaro, Albert P. Oloo Okoth, Isaac .Owino https://repository.maseno.ac.ke/handle/123456789/6060 2024-03-25T12:54:26Z 2024-01-09T00:00:00Z

Bijections for classes of labelled trees Nyariaro, Albert P. Oloo; Okoth, Isaac .Owino Trees are acyclic connected graphs. Plane trees, d-ary trees, binary trees, non crossing trees and their generalizations, which are families of trees, have been enumerated by many authors using various statistics. These trees are known to be enumerated by Catalan or Catalan-like formulas (Fuss-Catalan numbers). One of the most common approaches to the enumeration of these trees is by means of generating functions. Another method that can be used to enumerate them is by constructing bijections between sets of the same cardinality. The bijective method is preferred to other methods by many combinatorialists. So, in this paper, we construct bijections relating k-plane trees, k-noncrossing increasing trees, k-noncrossing trees, k-binary trees and weakly labelled k-trees.

2024-01-09T00:00:00Z Onyango, Christopher Amolo Okoth, Isaac Owino Kasyoki, Donnie Munyao https://repository.maseno.ac.ke/handle/123456789/5849 2023-11-15T17:49:58Z 2023-08-26T00:00:00Z

Enumeration of plane and d-ary tree-like structures Onyango, Christopher Amolo; Okoth, Isaac Owino; Kasyoki, Donnie Munyao Trees are generalized using various approaches such as considering tree-like structures. Some of the tree-like structures are Husimi graphs, cacti and oriented cacti. These graphs have been enumerated according to number of vertices, blocks, block types and degree sequences. Noncrossing and plane counterparts have also been enumerated by number of vertices, blocks and block types. In this paper, we enumerate plane Husimi graphs, cacti and oriented cacti according to root degree, outdegree of a given vertex and outdegree sequence. The d-ary tree like structures are also introduced in this paper and enumerated according to number of vertices, blocks, block types, outdegree sequence and number of leaves.

2023-08-26T00:00:00Z Abayo, Sylvester Arthur Okoth, Isaac Owino Kasyoki, Donnie Munyao https://repository.maseno.ac.ke/handle/123456789/5848 2023-11-15T17:43:23Z 2023-10-01T00:00:00Z

Reachability in complete t-ary trees Abayo, Sylvester Arthur; Okoth, Isaac Owino; Kasyoki, Donnie Munyao Mathematical trees such as Cayley trees, plane trees, binary trees, noncrossing trees, t-ary trees among others have been studied extensively. Reachability of vertices as a statistic has been studied in Cayley trees, plane trees, noncrossing trees and recently in t-ary trees where all edges are oriented from vertices of lower label towards vertices of higher label. In this paper, we obtain closed formulas as well as asymptotic formulas for the number of complete t-ary trees in which there are paths of a given length such that the terminal vertex is a sink, leaf sink, first child and non-first child. We also obtain number of trees in which there is a leftmost path of a given length.

2023-10-01T00:00:00Z Odongo, Benard A Manyonge, Alfred W Owego, Dancun O Opiyo, Richard O https://repository.maseno.ac.ke/handle/123456789/5795 2023-09-25T18:24:04Z 2023-09-15T00:00:00Z

On the Numerical Solution of Boundary Value Problem (BVP) of the Ordinary Differential Equation (ODE) - The Case of Steady-State Bio-Heat Equation with Combined Heat Transfer Coefficient by Pseudo-Spectral Collocation Method Odongo, Benard A; Manyonge, Alfred W; Owego, Dancun O; Opiyo, Richard O Spectral methods for the solution of a boundary value problem of an ordinary differential equation are reviewed with particular emphasis laid on pseudo-spectral collocation method. The pseudo-collocation method is then used to solve the one dimensional bio-heat equation with metabolic heat generation in cylindrical coordinates applied to human tissue. It was noticed that an increase in heat transfer coefficient (hA), enhanced the temperature but a decrease in the tissue thickness was observed when this coefficient was increased. The effects of the combined heat transfer coefficient are analyzed and the results indicate that the obtained solution can be used in the study of the thermal behaviour of a biological system with the potential to locate tumours in the living tissue.

2023-09-15T00:00:00Z Owino, Isaac. Okoth https://repository.maseno.ac.ke/handle/123456789/5424 2022-10-22T15:24:51Z 2022-01-01T00:00:00Z

Bijections of k-plane trees Owino, Isaac. Okoth A k-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set {1, 2, . . . , k}, the children of all vertices are ordered, and if (i, j) is an edge in the tree, where i and j are labels of adjacent vertices in the tree, then i + j ≤ k + 1. In this paper, we construct bijections between these trees and the sets of k-noncrossing increasing trees, locally oriented (k − 1)-noncrossing trees, Dyck paths, and some restricted lattice paths. https://pisrt.org/psrpress/j/odam/2022/1/bijections-of-k-plane-trees.pdf

2022-01-01T00:00:00Z