We describe and study the interplay between intuition and the precise description of graph algorithms. We learn how to use ordered rooted trees to develop and describe complex recursive algorithms. Depth first search, lineal spanning trees, chords, and backedges are used to understand complex algorithms such as those for biconnectivity and planarity. More general algoithms such as orderly algorithms are discussed in detail. We focus on the visualization of algorithms through the use of graphical and pictorial methods. This approach is both fun and powerful, preparing you to invent your own algorithms for a wide range of problems.