Bestfirst search
Bestfirst search is a search algorithm which explores a graph by expanding the most promising node chosen according to a specified rule.
Graph and tree search algorithms 

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Judea Pearl described bestfirst search as estimating the promise of node n by a "heuristic evaluation function which, in general, may depend on the description of n, the description of the goal, the information gathered by the search up to that point, and most important, on any extra knowledge about the problem domain."[1][2]
Some authors have used "bestfirst search" to refer specifically to a search with a heuristic that attempts to predict how close the end of a path is to a solution, so that paths which are judged to be closer to a solution are extended first. This specific type of search is called greedy bestfirst search[2] or pure heuristic search.[3]
Efficient selection of the current best candidate for extension is typically implemented using a priority queue.
The A* search algorithm is an example of a bestfirst search algorithm, as is B*. Bestfirst algorithms are often used for path finding in combinatorial search. Neither A* nor B* is a greedy bestfirst search, as they incorporate the distance from the start in addition to estimated distances to the goal.
Greedy BFS
Using a greedy algorithm, expand the first successor of the parent. After a successor is generated:[4]
 If the successor's heuristic is better than its parent, the successor is set at the front of the queue (with the parent reinserted directly behind it), and the loop restarts.
 Else, the successor is inserted into the queue (in a location determined by its heuristic value). The procedure will evaluate the remaining successors (if any) of the parent.
References
 Pearl, J. Heuristics: Intelligent Search Strategies for Computer Problem Solving. AddisonWesley, 1984. p. 48.
 Russell, Stuart J.; Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0137903952. pp. 94 and 95 (note 3).
 Korf, Richard E. (1999). "Artificial intelligence search algorithms". In Atallah, Mikhail J. (ed.). Handbook of Algorithms and Theory of Computation. CRC Press. ISBN 0849326494.
 https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume28/coles07ahtml/node11.html#modifiedbestfs Greedy BestFirst Search when EHC Fails, Carnegie Mellon