Maximal path length of binary trees

Cameron, Helen; Wood, Derick

doi:10.1016/0166-218x(94)90034-5

Cited by 8 publications

(9 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The notion of the Sackin index given by Definition 2, i.e. the sum of all path lengths from the root to any leaf ("external node") of the tree, is known as total external path length in computer science [3,15,16,28], where it is typically applied to ordered search trees. We will consider these more in-depth in Sect.…”

Section: Preliminariesmentioning

confidence: 99%

Extremal Values of the Sackin Tree Balance Index

Fischer¹

2021

Ann. Comb.

View full text Add to dashboard Cite

Tree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Also, concerning ordered search trees, more balanced ones allow for more efficient data structuring than imbalanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have only been provided for some of them, and only in the context of ordered binary (search) trees, not for general rooted trees. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves has been completely unknown. In this manuscript, we extend the findings on trees with minimal and maximal Sackin indices from the literature on ordered trees and subsequently use our results to provide formulas to explicitly calculate the numbers of such trees. We also extend previous studies by analyzing the case when the underlying trees need not be binary. Finally, we use our results to contribute both to the phylogenetic as well as the computer scientific literature using the new findings on Sackin minimal and maximal trees to derive formulas to calculate the number of both minimal and maximal phylogenetic trees as well as minimal and maximal ordered trees both in the binary and non-binary settings. All our results have been implemented in the Mathematica package SackinMinimizer, which has been made publicly available.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Extremal Values of the Sackin Tree Balance Index

Fischer¹

2021

Ann. Comb.

View full text Add to dashboard Cite

show abstract

“…The characterization of the maximum-path-length binary trees, for all sizes and fringe thicknesses, is still unsolved. Cameron [Cam91] and Cameron and Wood [CW94] give a partial solution by characterizing the maximumpath-length binary trees, for all sizes, fringe thicknesses, and heights, The détermination of the heights that guarantee maximum path length is the crucial unsolved problem.…”

Section: Discussionmentioning

confidence: 99%

“…Driven by similar concerns, we want to characterize the minimum path length trees for each size and fringe thickness. (The corresponding problem for maximum path length trees is still open; Klein and Wood [KW89] and Cameron and Wood [Cam91,CW94] have obtained partial results.) Recently, De Santis and Persiano [DP94] derived an attainable lower bound for the path length of binary trees of a given size and fringe thickness, when the fringe thickness is less than half of the size.…”

Section: Tv (Tv+ 3)mentioning

confidence: 99%

Binary trees, fringe thickness and minimum path length

Cameron¹,

Wood²

1995

RAIRO-Theor. Inf. Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We have used the characterization of the greedy représentations in the P2 number System in Cameron [Cam91] and Cameron and Wood [CW93] to establish an upper bound resuit for a class of binary trees. Every binary tree can be viewed as a perfect binary tree (a binary tree whose leaves ail appear on one le vel; see fig.…”

Section: Repeatedly We Set a I +-\ N/u^ Nmentioning

confidence: 99%

Pm numbers, ambiguity, and regularity

Cameron¹,

Wood²

1993

RAIRO-Theor. Inf. Appl.

View full text Add to dashboard Cite

L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Informatique théorique et Applications/Theoretical Informaties and Applications (vol. 27, n° 3, 1993, p. 261 à 275) Pm NUMBERS, AMBIGUITY, AND REGULARITY (*) (x) by H. A. CAMERON (2) and D. WOOD (2) Communicated by C. CHOFFRUT Abstract.-We introducé the pseudo-m-ray (Pm) number system, in which syms of the form J^ <2 £ (m I+1-l)/(m-1) are the représentations ofnumbers. We characterize the Pm représentations that are produced by the greedy algorithm and show that they form a regular set. In addition, we show that the set of Pm représentations that are the sole représentations for their corresponding numbers is also a regular set. Résumé.-Nous introduisons le système de numérotation pseudo-m-aire 0P m), dans lequel les sommes de la f orme £ ûj(m I+1)/(m-1) sont les représentations des nombres. Nous caractérisons les représentations de P m qui sont obtenues par l'algorithme vorace et nous montrons qu'elles forment un langage rationnel. De plus^ nous montrons que l'ensemble des représentations de P m des nombres qui ont une unique représentation, est un langage rationnel.

show abstract

Maximal path length of binary trees

Cited by 8 publications

References 2 publications

Extremal Values of the Sackin Tree Balance Index

Extremal Values of the Sackin Tree Balance Index

Binary trees, fringe thickness and minimum path length

Pm numbers, ambiguity, and regularity

Contact Info

Product

Resources

About