Tree terminology is far from standardized as many texts use the same terms inconsistently, but I can share what I have become accustomed to. I tend to use "tree" related to graph-theory terms, thus we have: tree: connected graph with no closed cycles. rooted tree: tree with a distinguished node called the root. Thus the "trees" you're referring to are actually rooted trees. Note that: o o | /|\ o and o o o / \ o o are the same tree but different rooted trees. Rooted trees correspond to the hierarchies used in data structures and other applications. Rooted tree is the name of the "X" in your question. We are often interested in applying some sort of combinatorial arrangement to the children of a node. A book I've referenced a lot on the subject _Combinatorial Species and Tree-Like Structures_ by Bergeron, Labelle and Leroux uses the term R-enriched rooted tree for such a structure where R represents some combinatorial structure. If R is a set we have rooted tree or what you're calling "X" If R is linear order we have ordered rooted tree or what you're calling "tree" If R is cycle we have mobile. ------ Original Message ------ Received: Wed, 11 Jul 2012 10:44:21 PM PDT From: Marc LeBrun <mlb@well.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Tree terminology
Speaking of terminology, I¹m drawing a blank:
Tree is to Composition AS X is to Partition
X == ...?
For example the following are distinct Trees, but equivalent X¹s:
O O / \ / \ O O O O / \ \ / / \ O O O O O O
Thanks! --MLB
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