  Graph searches and their respective search trees are widely used in
algorithmic graph theory. The problem whether a given spanning tree can be a
graph search tree has been considered for different searches, graph classes and
search tree paradigms. Similarly, the question whether a particular vertex can
be visited last by some search has been studied extensively in recent years. We
combine these two problems by considering the question whether a vertex can be
a leaf of a graph search tree. We show that for particular search trees,
including DFS trees, this problem is easy if we allow the leaf to be the first
vertex of the search ordering. We contrast this result by showing that the
problem becomes hard for many searches, including DFS and BFS, if we forbid the
leaf to be the first vertex. Additionally, we present several structural and
algorithmic results for search tree leaves of chordal graphs.
