The root of a tree is special, in that it has no parents. The leaves are special in that they have no children. The other nodes each have exactly one parent and more than zero children. Is there a word for that third kind? (Myself, I have been calling them "middle" or "tree-middle".)
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branches, or trunk if connected to the root, and twigs if connected to a leaf. – JMP Jul 29 '15 at 05:26
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1@JonMarkPerry: No, branches are the edges. – user21820 Jul 29 '15 at 05:28
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"Nodes that are neither trees nor leaves" or "Nodes with parents and children" – msinghal Jul 29 '15 at 05:29
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@user21820; you're right;i was thinking of the dual graph... – JMP Jul 29 '15 at 05:30
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1@MihirSinghal: Haha "neither roots nor leaves". – user21820 Jul 29 '15 at 05:31
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what if it's autumn? – JMP Jul 29 '15 at 05:32
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@JonMarkPerry: Then leaves leave and twigs become terminal. – user21820 Jul 29 '15 at 05:33
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1If I am discussing a specific vertex and it is of degree greater than two, I often call it a "branching point." Otherwise, "interior vertex" is sufficient to distinguish the vertex from leaves. A root may also be an interior vertex. – JMoravitz Jul 29 '15 at 05:34
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@JonMarkPerry: It doesn't, even in autumn. It just ends there. =D – user21820 Jul 29 '15 at 05:34
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@user21820; by definition an edge has exactly 2 vertices – JMP Jul 29 '15 at 05:35
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@JMoravitz: Normally "interior vertex" is reserved for "vertices that are in the interior (of a polytope)" but I guess context may be enough. – user21820 Jul 29 '15 at 05:36
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@JonMarkPerry: Come on it was a joke. There are some extensions of the concept of graphs that allow edges not to end in vertices though. – user21820 Jul 29 '15 at 05:37
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@user21820; such as? – JMP Jul 29 '15 at 05:40
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@user21820; we could only know a graph up to radius r, so beyond that is unknown – JMP Jul 29 '15 at 05:46
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@JonMarkPerry: I really cannot remember the name now... I think it was to facilitate talking about partitioning a graph. – user21820 Jul 29 '15 at 05:51
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@user21820; a cut? https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem – JMP Jul 29 '15 at 05:53
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@JonMarkPerry: Not that. For example see http://math.stackexchange.com/a/651880/21820 where I prove that any planar graph can be partitioned into claws (each claw does not have the vertices at the ends of the three fingers) and two vertices. – user21820 Jul 29 '15 at 05:56
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Usually you just need the concept of "internal nodes", which are nodes that are not leaves, because usually there is no need to distinguish between the root node and other internal nodes. The reason is that the root is often an arbitrary choice, whereas the unrooted tree still has the same internal nodes.
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