Is root an intrinsic property of a given tree?(Given a tree, can you uniquely determine the root?) Can't any vertex of a tree be chosen as a root? Aren't all trees rooted in that case?
2026-03-25 23:16:40.1774480600
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Is a 'root' an intrinsic property of a tree
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NO; not necessarily a tree has a root.
A rooted tree is a tree in which one vertex has been designated the root.
This is just one common example of a pointed set (or rooted set). Given any non-empty set, any one of the elements can be chosen to be the root. The result is a rooted set. That is, a set with the extra structure of a root. Thus, there is one set, but many rooted sets with the same underlying set. Therefore, a root is not an intrinsic property of a set, but it is an intrinsic part of a rooted set.
Notice that this is much different than, for example, the center of a tree as described in MSE question 1874429 "The center of a tree is a vertex or an edge." which is unique for a given tree.