Tree search
Tree search is the basic algorithm in which we can change into other uninformed and informed searches based on what methods we use to decide which candidate to expand.  Initially only the initial state is in the search tree as a simple.  The above figure used in the slides is the English version of the pseudo code version on pg 72.

In both, the current node is checked to see whether it is a goal state.  If so, the solution is returned (the path through the tree).  Otherwise the node is expanded and its descendents placed into the fringe in some order.  The specific order is important and subject to the strategy employed by the agent (e.g., DFS, BFS, etc.).
Note that the ordering the nodes in the fringe can be done in the REMOVE-FIRST, INSERT-ALL stages.  The EXPAND function itself cannot do this entirely by itself as it does not have access to the fringe, merely the set of successors that it generates.