Splaying

In ordinary bottom-up splaying, locate the selected node by a binary search tree search and repeatedly move it toward the root. If its parent is the root, perform a zig step. Otherwise use a zig-zig step when the node and its parent occupy the same left/right directions, or a zig-zag step when their directions differ. Each tree rotation preserves the inorder traversal and hence the represented key order. The process ends with the selected key at the root. Unsuccessful access policies can instead move the last visited node to the root; arguments about successful calls should say the requested key exists.

Every node of a splay tree with at least four distinct keys can become a leaf of a rooted tree using only successful splaying operations. At an extreme key, access that key and then its sole adjacent stored key; the consecutive keys in a binary search tree property makes the extreme key a leaf. For an interior key $x$ with two larger stored keys, take consecutive $a<x<b<c$ and access $c,a,b$; the mirror construction handles two smaller keys.

For a quick shape proof, use $k(L,R)$ for a node and its two subtrees. After accessing $c,a$, the right subtree of $a$ can only be $c(b(x,0),D)$, $c(x(0,b),D)$, $b(x,c(0,D))$, or $x(0,c(b,D))$, with outside subtrees $A$ below $a$ and $D$ above $c$. This follows by inspecting the final zig step, zig-zig step or zig-zag step that moves $a$ above the former root $c$. In each case, accessing $b$ yields $b(a(A,x),c(0,D))$, where $x$ is a leaf and $0$ denotes an empty subtree. These rotations prove the bound for arbitrary outside subtrees.

When a node and its parent occupy opposite left/right directions, rotate the selected node above its parent and then above its grandparent. These two tree rotations move the selected node two levels upward while preserving the binary search tree ordering.

When a node and its parent are both left children, first apply a right tree rotation to the grandparent, moving the parent upward, and then another right tree rotation moving the selected node upward. Use the left-right mirror image when both are right children. The order of these two tree rotations distinguishes the operation from repeatedly rotating only the selected node.

A single tree rotation moving a node above its parent when that parent is the root. A left child uses a right tree rotation, and a right child uses a left tree rotation.