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Depending on the problem at hand, pre-order, post-order, and especially one of the number of subtrees − 1 in-order operations may be optional. Also, in practice more than one of pre-order, post-order, and in-order operations may be required. For example, when inserting into a ternary tree, a pre-order operation is performed by comparing items.
"A binary tree is threaded by making all right child pointers that would normally be null point to the in-order successor of the node (if it exists), and all left child pointers that would normally be null point to the in-order predecessor of the node." [1] This assumes the traversal order is the same as in-order traversal of the tree. However ...
Remove the root of a tree and process each of its children, or; Join two trees together by making one tree a child of the other. Operation (1) it is very efficient. In LCRS representation, it organizes the tree to have a right child because it does not have a sibling, so it is easy to remove the root. Operation (2) it is also efficient.
However, if a tree is defined in terms of nodes, and list of trees - something like ... e.g data Tree a = Node a | (Node a, [ Tree a]) .... you might prefer data Tree a = EmptyTree| (Node a, [ Tree a]) then the generalisation would be to recursively visit the node first, followed by each tree in the list in list order.
Often trees have a fixed (more properly, bounded) branching factor , particularly always having two child nodes (possibly empty, hence at most two non-empty child nodes), hence a "binary tree". Allowing empty trees makes some definitions simpler, some more complicated: a rooted tree must be non-empty, hence if empty trees are allowed the above ...
The two addends correspond to the two data constructors for a list. The Nil data constructor, a sentinel node that holds no data, is represented by the unit type 1 {\displaystyle 1} , and the Cons constructor is represented as a product of the head element X {\displaystyle X} and the tail L {\displaystyle L} of the list.
The Nested Set model is appropriate where the tree element and one or two attributes are the only data, but is a poor choice when more complex relational data exists for the elements in the tree. Given an arbitrary starting depth for a category of 'Vehicles' and a child of 'Cars' with a child of 'Mercedes', a foreign key table relationship must ...
Then one endpoint of edge e is in set V and the other is not. Since tree Y 1 is a spanning tree of graph P, there is a path in tree Y 1 joining the two endpoints. As one travels along the path, one must encounter an edge f joining a vertex in set V to one that is not in set V.