Editing Wedderburn–Etherington number

{{Short description|Number that can be used to count certain kinds of binary trees}}
In [[mathematics]] and [[computer science]], the '''Wedderburn–Etherington numbers''' are an [[integer sequence]] named after [[Ivor Malcolm Haddon Etherington]]<ref name="e37">{{citation
 | last = Etherington | first = I. M. H. | author-link = Ivor Malcolm Haddon Etherington
 | doi = 10.2307/3605743
 | issue = 242
 | journal = [[Mathematical Gazette]]
 | pages = 36–39, 153
 | title = Non-associate powers and a functional equation
 | volume = 21
 | year = 1937| jstor = 3605743 | s2cid = 126360684 }}.</ref><ref name="e39">{{citation
 | last = Etherington | first = I. M. H. | author-link = Ivor Malcolm Haddon Etherington
 | issue = 2
 | journal = Proc. R. Soc. Edinburgh
 | pages = 153–162
 | title = On non-associative combinations
 | volume = 59
 | year = 1939| doi = 10.1017/S0370164600012244 }}.</ref> and [[Joseph Wedderburn]]<ref name="w">{{citation
 | last = Wedderburn | first = J. H. M. | author-link = Joseph Wedderburn
 | doi = 10.2307/1967710
 | issue = 2
 | journal = [[Annals of Mathematics]]
 | pages = 121–140
 | title = The functional equation <math>g(x^2) = 2ax + [g(x)]^2</math>
 | volume = 24
 | year = 1923| jstor = 1967710 }}.</ref> that can be used to count certain kinds of [[binary tree]]s. The first few numbers in the sequence are
:0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, ... ({{OEIS2C|A001190}})

==Combinatorial interpretation==
[[File:Wedderburn-Etherington trees.svg|thumb|360px|Otter trees and weakly binary trees, two types of rooted binary tree counted by the Wedderburn–Etherington numbers]]
These numbers can be used to solve several problems in [[combinatorial enumeration]]. The ''n''th number in the sequence (starting with the number 0 for ''n''&nbsp;=&nbsp;0)
counts
*The number of unordered [[rooted tree]]s with ''n'' leaves in which all nodes including the root have either zero or exactly two children.<ref name="oeis">{{Cite OEIS|A001190}}.</ref> These trees have been called Otter trees,<ref>{{citation
 | last1 = Bóna | first1 = Miklós | author1-link = Miklós Bóna
 | last2 = Flajolet | first2 = Philippe | author2-link = Philippe Flajolet
 | arxiv = 0901.0696
 | doi = 10.1239/jap/1261670685
 | issue = 4
 | journal = Journal of Applied Probability
 | mr = 2582703
 | pages = 1005–1019
 | title = Isomorphism and symmetries in random phylogenetic trees
 | volume = 46
 | year = 2009| bibcode = 2009arXiv0901.0696B | s2cid = 5452364 }}.</ref> after the work of Richard Otter on their combinatorial enumeration.<ref>{{citation
 | last = Otter | first = Richard
 | doi = 10.2307/1969046
 | journal = [[Annals of Mathematics]]
 | mr = 0025715
 | pages = 583–599
 | series = Second Series
 | title = The number of trees
 | volume = 49
 | issue = 3
 | year = 1948| jstor = 1969046
 }}.</ref> They can also be interpreted as unlabeled and unranked [[dendrogram]]s with the given number of leaves.<ref name="dendrogram">{{citation
 | last = Murtagh | first = Fionn
 | doi = 10.1016/0166-218X(84)90066-0
 | issue = 2
 | journal = [[Discrete Applied Mathematics]]
 | mr = 727923
 | pages = 191–199
 | title = Counting dendrograms: a survey
 | volume = 7
 | year = 1984| doi-access = free
 }}.</ref>
*The number of unordered rooted trees with ''n'' nodes in which the root has degree zero or one and all other nodes have at most two children.<ref name="oeis"/> Trees in which the root has at most one child are called planted trees, and the additional condition that the other nodes have at most two children defines the weakly binary trees. In [[chemical graph theory]], these trees can be interpreted as [[isomer]]s of [[polyene]]s with a designated leaf atom chosen as the root.<ref>{{citation
 | last1 = Cyvin | first1 = S. J.
 | last2 = Brunvoll | first2 = J.
 | last3 = Cyvin | first3 = B.N.
 | doi = 10.1016/0166-1280(95)04329-6
 | issue = 3
 | journal = Journal of Molecular Structure: THEOCHEM
 | pages = 255–261
 | title = Enumeration of constitutional isomers of polyenes
 | volume = 357
 | year = 1995}}.</ref>
*The number of different ways of organizing a [[single-elimination tournament]] for ''n'' players (with the player names left blank, prior to seeding players into the tournament).<ref>{{citation
 | last = Maurer | first = Willi
 | journal = The Annals of Statistics
 | jstor = 2958441
 | mr = 0371712
 | pages = 717–727
 | title = On most effective tournament plans with fewer games than competitors
 | volume = 3
 | issue = 3
 | year = 1975
 | doi = 10.1214/aos/1176343135| doi-access = free
 }}.</ref> The pairings of such a tournament may be described by an Otter tree.
*The number of different results that could be generated by different ways of grouping the expression <math>x^n</math> for a binary multiplication operation that is assumed to be [[commutative]] but neither [[associative]] nor [[idempotent]].<ref name="oeis"/> For instance <math>x^5</math> can be grouped into binary multiplications in three ways, as <math>x(x(x(xx)))</math>, <math>x((xx)(xx))</math>, or <math>(xx)(x(xx))</math>. This was the interpretation originally considered by both Etherington<ref name="e37"/><ref name="e39"/> and Wedderburn.<ref name="w"/> An Otter tree can be interpreted as a grouped expression in which each leaf node corresponds to one of the copies of <math>x</math> and each non-leaf node corresponds to a multiplication operation. In the other direction, the set of all Otter trees, with a binary multiplication operation that combines two trees by making them the two subtrees of a new root node, can be interpreted as the [[Free object|free]] [[commutative magma]] on one generator <math>x</math> (the tree with one node). In this algebraic structure, each grouping of <math>x^n</math> has as its value one of the ''n''-leaf Otter trees.<ref>This equivalence between trees and elements of the free commutative magma on one generator is stated to be "well known and easy to see" by {{citation
 | last = Rosenberg | first = I. G.
 | doi = 10.1016/0166-218X(86)90068-5
 | issue = 1
 | journal = [[Discrete Applied Mathematics]]
 | mr = 829338
 | pages = 41–59
 | title = Structural rigidity. II. Almost infinitesimally rigid bar frameworks
 | volume = 13
 | year = 1986| doi-access = free
 }}.</ref>

==Formula==
The Wedderburn–Etherington numbers may be calculated using the [[recurrence relation]]
:<math>a_{2n-1}=\sum_{i=1}^{n-1} a_i a_{2n-i-1}</math>
:<math>a_{2n}=\frac{a_n(a_n+1)}{2}+\sum_{i=1}^{n-1} a_i  a_{2n-i}</math>
beginning with the base case <math>a_1=1</math>.<ref name="oeis"/>

In terms of the interpretation of these numbers as counting rooted binary trees with ''n'' leaves, the summation in the recurrence counts the different ways of partitioning these leaves into two subsets, and of forming a subtree having each subset as its leaves. The formula for even values of ''n'' is slightly more complicated than the formula for odd values in order to avoid double counting trees with the same number of leaves in both subtrees.<ref name="dendrogram"/>

==Growth rate==
The Wedderburn–Etherington numbers grow [[asymptotic analysis|asymptotically]] as
:<math>a_n \approx \sqrt{\frac{\rho+\rho^2B'(\rho^2)}{2\pi}} \frac{\rho^{-n}}{n^{3/2}},</math>
where ''B'' is the [[generating function]] of the numbers and ''ρ'' is its [[radius of convergence]], approximately 0.4027 {{OEIS|A240943}}, and where the constant given by the part of the expression in the square root is approximately 0.3188 {{OEIS|A245651}}.<ref>{{citation
 | last = Landau | first = B. V.
 | issue = 2
 | journal = [[Mathematika]]
 | mr = 0498168
 | pages = 262–265
 | title = An asymptotic expansion for the Wedderburn-Etherington sequence
 | volume = 24
 | year = 1977
 | doi=10.1112/s0025579300009177}}.</ref>

==Applications==
{{harvtxt|Young|Yung|2003}} use the Wedderburn–Etherington numbers as part of a design for an [[encryption]] system containing a hidden [[Backdoor (computing)|backdoor]]. When an input to be encrypted by their system can be sufficiently [[data compression|compressed]] by [[Huffman coding]], it is replaced by the compressed form together with additional information that leaks key data to the attacker. In this system, the shape of the Huffman coding tree is described as an Otter tree and encoded as a binary number in the interval from 0 to the  Wedderburn–Etherington number for the number of symbols in the code. In this way, the encoding uses a very small number of bits, the base-2 logarithm of the Wedderburn–Etherington number.<ref>{{citation
 | last1 = Young | first1 = Adam
 | last2 = Yung | first2 = Moti | author2-link = Moti Yung
 | contribution = Backdoor attacks on black-box ciphers exploiting low-entropy plaintexts
 | doi = 10.1007/3-540-45067-X_26
 | isbn = 978-3-540-40515-3
 | pages = 297–311
 | publisher = Springer
 | series = [[Lecture Notes in Computer Science]]
 | title = Proceedings of the 8th Australasian Conference on Information Security and Privacy (ACISP'03)
 | volume = 2727
 | year = 2003}}.</ref>

{{harvtxt|Farzan|Munro|2008}} describe a similar encoding technique for rooted unordered binary trees, based on partitioning the trees into small subtrees and encoding each subtree as a number bounded by the Wedderburn–Etherington number for its size. Their scheme allows these trees to be encoded in a number of bits that is close to the information-theoretic lower bound (the base-2 logarithm of the Wedderburn–Etherington number) while still allowing constant-time navigation operations within the tree.<ref>{{citation
 | last1 = Farzan | first1 = Arash
 | last2 = Munro | first2 = J. Ian | author2-link = Ian Munro (computer scientist)
 | contribution = A uniform approach towards succinct representation of trees
 | doi = 10.1007/978-3-540-69903-3_17
 | mr = 2497008
 | pages = 173–184
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Algorithm theory—SWAT 2008
 | volume = 5124
 | year = 2008| isbn = 978-3-540-69900-2
 }}.</ref>

{{harvtxt|Iserles|Nørsett|1999}} use unordered binary trees, and the fact that the Wedderburn–Etherington numbers are significantly smaller than the numbers that count ordered binary trees, to significantly reduce the number of terms in a series representation of the solution to certain [[differential equation]]s.<ref>{{citation
 | last1 = Iserles | first1 = A.
 | last2 = Nørsett | first2 = S. P.
 | doi = 10.1098/rsta.1999.0362
 | issue = 1754
 | journal =  Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
 | mr = 1694700
 | pages = 983–1019
 | title = On the solution of linear differential equations in Lie groups
 | volume = 357
 | year = 1999| bibcode = 1999RSPTA.357..983I
 | s2cid = 90949835
 | url = https://cds.cern.ch/record/323789
 }}.</ref>

==See also==
* [[Catalan number]]
* [[Cryptography]]
* [[Information theory]]

==References==
{{reflist}}

== Further reading==
*{{citation
 | last = Finch
 | first = Steven R.
 | doi = 10.1017/CBO9780511550447
 | isbn = 978-0-521-81805-6
 | location = Cambridge
 | mr = 2003519
 | pages = [https://archive.org/details/mathematicalcons0000finc/page/295 295–316]
 | publisher = Cambridge University Press
 | series = Encyclopedia of Mathematics and its Applications
 | title = Mathematical constants
 | volume = 94
 | year = 2003
 | url = https://archive.org/details/mathematicalcons0000finc/page/295
 }}.

{{Classes of natural numbers}}

{{DEFAULTSORT:Wedderburn-Etherington number}}
[[Category:Integer sequences]]
[[Category:Trees (graph theory)]]
[[Category:Graph enumeration]]