Searching nodes in semi-splay tree
To complete the search process for an element $x$ that is not in the tree, you have to go all the way to a leaf node to confirm that $x$ is not in the tree.  If you repeatedly search for such an element $x$, you have to splay on an unsuccessful search, otherwise the query time for $x$ could indefinitely remain as bad as $\Theta(n)$, even in an amortized sense.