https://mathworld.wolfram.com/RootedTree.html: “A rooted tree is a tree in which a special ("labeled") node is singled out. This node is called the "root" or (less commonly) "eve" of the tree. Rooted trees are equivalent to oriented trees (Knuth 1997, pp. 385-399). A tree which is not rooted is sometimes called a free tree, although the unqualified term "tree" generally refers to a free tree.”
> A tree which is not rooted is sometimes called a free tree, although the unqualified term "tree" generally refers to a free tree.
I wonder if that was once the case but no longer is. I'm learning I think of trees mostly through the lens of data structures and not graph theory and I imagine more people do than not.
You gotta be careful -- a path does not exist between two leaf nodes, unless your edges are non-directional, in which case you're right. Oh and restrict it to 'only visit each node once' given undirected.
Visiting a node twice implies a loop, which implies two paths between your original nodes.
Let’s say we want to get from A to B, but we visit Y twice. That means there’s a segment of A~B that is Y~Y (a loop). However, since we’re talking about undirected edges, we can reverse that segment. Then both our original A~B and the A~B with Y~Y reversed are paths from A to B.
So if there’s a single path from A to B, there cannot be a Y we visit twice along that path.
(Please don’t take this as criticism; my pedantic nature got the better of me.)
Yes, but in an UndirectedGraph you can do loops easy peasy. A->b->a->b etc. if you have a tree with a parent node, then the tree (A{children:[b], parent: null}, B{children:[], parent: A}) still may allow for this behavior (graph representation unions parent and children since a parent in tree parlence is basically 'incoming-edges' in graph parlence while children is 'outgoing-edges')
I was trying to find terminology for something
Like 'all possible paths between these two nodes contain exactly one Hamiltonian sub-path' but I think that's a bit of a circular definition
All that said I'm running off 3 hours of sleep and about to recoup so that probably explains my non constructive comments here
I think that I shall never see; A graph more lovely than a tree. A tree whose crucial property; Is loop-free connectivity. - Radia Perlman, inventor of spanning tree
Slight nitpick: in mathematics, a tree need not have a root.
https://mathworld.wolfram.com/FreeTree.html: “a normal tree with no node singled out for special treatment”
https://mathworld.wolfram.com/RootedTree.html: “A rooted tree is a tree in which a special ("labeled") node is singled out. This node is called the "root" or (less commonly) "eve" of the tree. Rooted trees are equivalent to oriented trees (Knuth 1997, pp. 385-399). A tree which is not rooted is sometimes called a free tree, although the unqualified term "tree" generally refers to a free tree.”