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Hillclimbing proceeds as per values of each nodes.Whereas BFS follows edges of the tree.

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Q: Difference between hillclimbing and best first search?
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Difference between hill climbing and best first search?

hill climbing is a technique in which path is find form strating position to goal node....it finds a better solution by repeatedly changing a single element to the solution...on the other hand best first search finds the best way to find the goal....