Spanning tree (mathematics)

In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G, with minimum possible number of edges. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (but see Spanning forests below). If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T (that is, a tree has a unique spanning tree and it is itself).

Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem.

In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree.

The Internet and many other telecommunications networks have transmission links that connect nodes together in a mesh topology that includes some loops. In order to "avoid bridge loops and "routing loops", many routing protocols designed for such networks—including the , Open Shortest Path First, , Augmented tree-based routing, etc. -- require each router to remember a spanning tree.

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