Searching nodes in semi-splay tree If you search for a node in a semi-splay tree, it's basically to push certain nodes closer to the root, to reduce future search operations.

My course also says that if you search for a node and the node is not in the semi-splay tree, you still have to perform semi-play on the search path in the tree.
But isn't that basically pushing nodes to the root that aren't actually have been searched for?

So my question would be : why do you still have to perform semi-splay even if the node you search for is not in the 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.