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On the number of deepest nodes in ordered trees

R. Kemp
1990 Discrete Mathematics  
Let Qn.k" be the number of all n-node ordered trees with r nodes of maximum level k and let B"*" be the number of all r-tuply rooted ordered trees with n nodes and height less than or equal to k.  ...  Finally, assuming that all n-node trees with height k are equally likely, we show that such a tree has 4 deepest nodes on the average for fixed k.  ...  Acknowledgements The author likes to thank the referee for his detailed criticism and the suggestions for improving the clarity of the presentation.  ... 
doi:10.1016/0012-365x(90)90064-o fatcat:cabvzf4bxvgktenevsfbqrgzbq

Deepest nodes in marked ordered trees [article]

Helmut Prodinger
2022 arXiv   pre-print
We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches 5/3 when the number of nodes gets large.  ...  This is to be compared to standard ordered trees where the average number of deepest nodes approaches 2.  ...  We ignore the tree with one node and one deepest node.  ... 
arXiv:2202.05455v2 fatcat:kz723yn24rcvxk2alpzxwtdq2a


1995 International Journal of Foundations of Computer Science  
Our results improve a number of bounds, in particular the bounds on the expected number of nodes and the expected space utilization. We also study 2-3 trees built by using over ow techniques.  ...  We present a fourth-order fringe analysis for the expected behavior of 2-3 trees, which includes 97% of the elements in the tree.  ...  EZG + 82] for ways of computing each one of these measures from the fringe. One basic technique for concurrent access in 2-3 trees is to lock the deepest safe node in the insertion path KW80].  ... 
doi:10.1142/s0129054195000020 fatcat:jhdhrn3bujg53ebduw3ua2ewru

Fast Sequential Summation Algorithms Using Augmented Data Structures [article]

Vadim Stadnik
2014 arXiv   pre-print
both logarithmic running time and logarithmic computational cost in terms of the total number of operations.  ...  The important practical advantage of the fast sequential summation algorithms is that they do not require supercomputers and can run on the cheapest single processor systems.  ...  A skip list guarantees logarithmic running time of the discussed algorithms on average only.  ... 
arXiv:1404.1560v1 fatcat:5xpsed6z4va7fhd5dpm4viucua

Avoidance of multicast incapable branching nodes for multicast routing in WDM networks

Fen Zhou, Miklos Molnar, Bernard Cousin
2008 2008 33rd IEEE Conference on Local Computer Networks (LCN)  
the shortest path based multicast tree and maintains good parts of the shortest path tree to reduce the end-to-end delay.  ...  Although many multicast routing algorithms have been proposed in order to reduce the total cost in WDM optical networks, the link stress and delay are two parameters which are not always taken into consideration  ...  To keep one branch of MIB nodes in the shortest path tree, critical articulation and deepest branch heuristics are introduced.  ... 
doi:10.1109/lcn.2008.4664188 dblp:conf/lcn/ZhouMC08 fatcat:crs5kgqnpbbz3hwnhrhxwmv7by

The theory of fringe analysis and its application to 23 trees and b-trees

Bernhard Eisenbarth, Nivio Ziviani, Gaston H. Gonnet, Kurt Mehlhorn, Derick Wood
1982 Information and Control  
As a consequence, bounds and some exact results on the expected number of splits per insertion and on the expected depth of the deepest safe node in 2-3 trees and B-trees, on the expected height of 2-3  ...  trees are obtained, and improvements of the bounds on the expected number of nodes in 2-3 trees are derived also.  ...  Case (i) is obvious: the number of external nodes is equal to the number of keys in the tree plus one.  ... 
doi:10.1016/s0019-9958(82)90534-4 fatcat:d7mv7e3aurhrth25gfhw6ahrfi

Persistence of the Jordan center in Random Growing Trees [article]

Sarath Pattathil, Nikhil Karamchandani, Dhruti Shah
2018 arXiv   pre-print
In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models.  ...  persists on a single vertex after a finite number of timesteps.  ...  In each of these models, one new node is added to the tree at any time.  ... 
arXiv:1712.08712v2 fatcat:3xcq7cw3wze7da7oiblgtduc6a

A New Method for Concurrency in B-Trees

Yat-Sang Kwong, D. Wood
1982 IEEE Transactions on Software Engineering  
In this paper we study the problem of supporting concurrent operations in B-trees and their variants.  ...  In particular, we introduce the technique of side-branching in our new solution.  ...  The idea of bulk merging in the discussion on intranodal concurrency was suggested by the referee.  ... 
doi:10.1109/tse.1982.235251 fatcat:qxavw7b3mbhs5j5abz4nr4olda


2000 Biocomputing 2001  
(on the order of log N, where N is the number of taxa).  ...  In other words, the number of taxa in an analysis can be doubled and only an arithmetic increase in the number of characters is required to maintain the same level of accuracy.  ...  Acknowledgements We thank Byron Adams, Bob Kuzoff, Steve Nadler, Ashleigh Smythe, Phil Ward, and the UCD Phylogenetics Discussion group for comments.  ... 
doi:10.1142/9789814447362_0053 fatcat:bfvixtdbo5g4foz2ykd7lyww54

On-line construction of compact suffix vectors and maximal repeats

Élise Prieur, Thierry Lecroq
2008 Theoretical Computer Science  
Not only do we show that it is possible to directly build a compact suffix vector but we will also show that this on-line construction can be faster than the construction of the extended suffix vector.  ...  We propose an on-line linear algorithm for directly constructing a compact suffix vector.  ...  In a reduced box, we store the deepest node and the number of nodes represented in the box. The compaction method is presented in Fig. 3 .  ... 
doi:10.1016/j.tcs.2008.06.017 fatcat:f64iacdfvngffmais4ta5vpb3y


Khushbakov Sherzod, Khamraev Mansur, Bakhtiyorova Mokhiruy
2022 Zenodo  
Graph algorithms are used to solve the problems of representing graphs as networks like airline flights, how the Internet is connected, or social network connectivity on Facebook.  ...  This article will introduce you to the tools you need to master important graph algorithms and the optimal search algorithm in Python.  ...  is a graph in the form of a two-dimensional matrix with dimensions , where V is the number of graph vertices.  ... 
doi:10.5281/zenodo.6502326 fatcat:qq3o4lwxqvhpzitz4di3h37g64

Avoidance of multicast incapable branching nodes for multicast routing in WDM networks

Fen Zhou, Miklós Molnár, Bernard Cousin
2009 Photonic network communications  
Although many multicast routing algorithms have been proposed in order to reduce the total number of wavelength channels used (total cost) for a multicast session, the maximum number of wavelengths required  ...  nodes in the USA Longhaul topology, (2) critical articulation and deepest branch heuristics are used to process the MIB nodes, (3) a distance-based light-tree reconnection algorithm is proposed to create  ...  To keep one branch of MIB nodes in the shortest path tree, critical articulation and deepest branch heuristics are introduced.  ... 
doi:10.1007/s11107-009-0200-3 fatcat:t6cuabmchvecxij4u4dsxlftt4

Suffix Vector: A Space-Efficient Suffix Tree Representation [chapter]

Krisztián Monostori, Arkady Zaslavsky, István Vajk
2001 Lecture Notes in Computer Science  
The basic idea behind the representation is that we are storing the nodes of the tree along with the string itself, thus edge labels can directly be read from the string.  ...  We also show that our representation is not only better in terms of space but it is also faster to retrieve information from the tree.  ...  If the number of nodes value equals to the deepest node value it means that the depth of shortest node is one character. One-character-deep nodes do not need to store a suffix link.  ... 
doi:10.1007/3-540-45678-3_60 fatcat:yhajmjofibfirolnhqxsancktu

Substring Complexities on Run-length Compressed Strings [article]

Akiyoshi Kawamoto, Tomohiro I
2022 arXiv   pre-print
Let S_T(k) denote the set of distinct substrings of length k in a string T, then the k-th substring complexity is defined by its cardinality |S_T(k)|.  ...  In this paper, given T of length n in the run-length compressed form of size r, we show that δ can be computed in 𝐶_𝗌𝗈𝗋𝗍(r, n) time and O(r) space, where 𝐶_𝗌𝗈𝗋𝗍(r, n) = O(min (r r, r _r n)) is  ...  This work was supported by JSPS KAKENHI Grant Number 22K11907.  ... 
arXiv:2205.12421v1 fatcat:xdx3ix6l3jg3fdw6mjaebgohwa

A Suffix Tree Or Not a Suffix Tree? [chapter]

Tatiana Starikovskaya, Hjalte Wedel Vildhøj
2015 Lecture Notes in Computer Science  
In this paper we study the structure of suffix trees. Given an unlabeled tree τ on n nodes and suffix links of its internal nodes, we ask the question "Is τ a suffix tree?"  ...  Deciding if τ is a suffix tree is not an easy task, because, with no restrictions on the final symbol, we cannot guess the length of a string that realizes τ from the number of leaves.  ...  We define the parent par(x) of a node x to be the deepest explicit node on the path from the root to x.  ... 
doi:10.1007/978-3-319-19315-1_30 fatcat:bnckjfkztnavznoxjmmrmxhgre
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