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Henzinger and King [2] suggest to represent a given tree by keeping its Euler tour in a balanced binary search tree, keyed by the index in the tour. So for example, the unbalanced tree in the example above, having 7 nodes, will be represented by a balanced binary tree with 14 nodes, one for each time each node appears on the tour.
Finding the frustration index is an NP-hard problem. One can see the NP-hard complexity by observing that the frustration index of an all-negative signed graph is the same as the maximum cut problem in graph theory, which is NP-hard. The frustration index is important in a model of spin glasses, the mixed Ising model. In this model, the signed ...
A more general problem is to count spanning trees in an undirected graph, which is addressed by the matrix tree theorem. (Cayley's formula is the special case of spanning trees in a complete graph.) The similar problem of counting all the subtrees regardless of size is #P-complete in the general case (Jerrum (1994)).
Tree topology, a topology based on a hierarchy of nodes in a computer network; Tree diagram (physics), an acyclic Feynman diagram, pictorial representations of the mathematical expressions governing the behavior of subatomic particles; Outliners, a common software application that is used to generate tree diagrams; Network diagram; Tree ...
This traversal is guided by the comparison function. In this case, the node always replaces a NULL reference (left or right) of an external node in the tree i.e., the node is either made a left-child or a right-child of the external node. After this insertion, if a tree becomes unbalanced, only ancestors of the newly inserted node are unbalanced.
For example, the ordered tree on the left and the binary tree on the right correspond: An example of converting an n-ary tree to a binary tree. In the pictured binary tree, the black, left, edges represent first child, while the blue, right, edges represent next sibling. This representation is called a left-child right-sibling binary tree.
In the depicted unbalanced and balanced trees, the balancing of the leftmost 3-element subtree doesn't appear to be able to be done by tree rotations as they are defined on the tree rotations page. When the 9 is rotated out and the 14 in, the twelve will switch to the opposite side, maintaining the imbalance.
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.