Editing Wedderburn–Etherington number (section)

==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>