User blog:Hyp cos/extended Kruskal theorem

The term "extended Kruskal theorem" showed in Bird's introduction page of his array notation (page 27). It generates functions with growth rate \(\psi(\Omega_\omega)\), much stronger than the TREE function.

Definitions
(The definitions of embeddings and the order types are derived here.)

In a tree, let inf(x,y) be the nearest common ancestor of node x and y. So inf(x,y)=x iff x is ancestor of y or x=y. In labeled tree, let l(x) be the label of node x. Here we discuss trees with labels in positive integers.

Labeled tree A is strongly gap-embeddable into B if there exists an injection f from nodes of A to nodes of B such that: Also, labeled tree A is weakly gap-embeddable into B if there exists an injection f from nodes of A to nodes of B fitting conditions 1, 2 and 3. Labeled tree A is embeddable into B if there exists an injection f from nodes of A to nodes of B fitting conditions 1 and 2. Unlabeled tree A is embeddable into B if there exists an injection f from nodes of A to nodes of B fitting conditions 1.
 * 1) For any x and y of A, f(inf(x,y)) = inf(f(x),f(y))
 * 2) For any x of A, l(x) = l(f(x))
 * 3) For any x of A, for any child y of x, for any node t of B, if f(x) is ancestor of t and t is ancestor of f(y), then l(t) ≥ l(f(y))
 * 4) Let r be the root of A; for any t of B, if t is ancestor of f(r), then l(t) ≥ l(f(r))

Equivalently, these 4 kinds of embedding can be stated as follows: 4 functions are derived from these embeddings. In TREE function, labels are symmetric. But in wgap function or sgap function, labels are asymmetric; the size of label numbers matter.
 * (Unlabeled or labeled) tree A is embeddable into B if A can be obtained from B by finite applications of
 * deleting a leaf, or
 * deleting a node with exactly one child.
 * Labeled tree A is weakly gap-embeddable into B if A can be obtained from B by finite applications of
 * deleting a leaf, or
 * deleting the root with exactly one child, or
 * deleting a non-root node with exactly one child with lessequal label to it.
 * Labeled tree A is strongly gap-embeddable into B if A can be obtained from B by finite applications of
 * deleting a leaf, or
 * deleting a node with exactly one child with lessequal label to it.
 * tree(n) is the maximal length of sequence of unlabeled tree T1, T2, ... such that
 * For any i, Ti has at most n+i nodes.
 * For any i < j, Ti is not embeddable into Tj.
 * TREE(n) is the maximal length of sequence of {1, 2, ..., n}-labeled tree T1, T2, ... such that
 * For any i, Ti has at most i nodes.
 * For any i < j, Ti is not embeddable into Tj.
 * wgap(n) is the maximal length of sequence of {1, 2, ..., n}-labeled tree T1, T2, ... such that
 * For any i, Ti has at most i nodes.
 * For any i < j, Ti is not weakly gap-embeddable into Tj.
 * sgap(n) is the maximal length of sequence of {1, 2, ..., n}-labeled tree T1, T2, ... such that
 * For any i, Ti has at most i nodes.
 * For any i < j, Ti is not strongly gap-embeddable into Tj.

The growth rate of tree, TREE, wgap and sgap function are \(\psi(\Omega^{\Omega^\omega})\), \(\psi(\Omega^{\Omega^\omega\omega})\), \(\psi(\Omega_\omega)\) and \(\psi(\Omega_\omega)\) respectively.

tree and TREE
For ordering of tree and TREE sequences, and a sequence for TREE(3), see here.