In graph theory what is the difference between a rooted tree and a free tree ? What is normally meant when just the plain "tree" is used ?
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In graph theory, the basic definition of a tree is that it is a graph without cycles. This definition does not use any specific node as a root for the tree. A rooted tree introduces a parent — child relationship between the nodes and the notion of depth in the tree.
Roughly and visually, adding a root to a tree just corresponds to grabbing one of the nodes of the tree and hanging the tree with this node. Once the tree is hanged, each node has a depth related to its height, a parent to which it is suspended and several children which hang from this node.
A rooted tree comes with one of its vertices specially designated to be the "root" node, such that there's an implicit notion of "towards the root" and "away from the root" for each edge.
In a free tree there's no designated root vertex.
You can make a free tree into a rooted one by choosing any of its vertices to be the root.
In my knowledge, in rooted tree, there is one special vertex, root. Whereas any vertex in a free tree can be considered as root.