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Kruskal, Prims and reverse-delete algorithms to find the Minimum Spanning Tree in a graph, are fun, not that hard, and very practical. http://en.wikipedia.org/wiki/Minimum_spanning_tree

Think about it, every time you get a google map direction/route, one of them is in play.



Also a bit different, but perhaps even more widespread application of spanning trees is STP family of L2 network bridging protocols.

http://en.wikipedia.org/wiki/Rapid_Spanning_Tree_Protocol




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