Splay trees: why are depths of nodes on the access path halved?
Nobody has answered this question yet, and I can give some hints how to solve it.
I would much rather post this as a comment, but I can't since I don't have enough reputation.
So here is a link to the original paper: https://www.cs.cmu.edu/~sleator/papers/self-adjusting.pdf
The basic idea is looking at zig-zig and zig-zag (zig only happens once when the parent of x is the root!):


Notice that if a node was in the search path before x was pushed to that position, it is now either in A or in B.
Notice that all nodes in A and B have at least 1 depth less than before the zig-zig operation



Notice that if a node was in the search path before x was pushed to that position, it is now either in B or in C.
Notice that all nodes in B and C have at least 1 depth less than before the zig-zag operation

Now just think about how many zig-zig and zig-zag operations there are for a node with depth d in the search path.