User:King2218/Hydras

Simple Hydra
Say we have a rooted tree with a finite number of nodes. After we chop off a leaf A, we consider the following rules:


 * 1) Call the parent of the leaf B. If B is the root, we simply do nothing.
 * 2) If B is not the root, call B's parent C. Take the subtree that contains B and all of its descendants. Copy that subtree and attach the new subtree to C.

SH(n) is the largest number of steps to reduce a hydra with length n to the root.

Values
SH(1) = 1

SH(2) = 3

SH(3) = 8

SH(4) = 38

SH(5) = approx. 33500 (I lost count on this one)

Generalized Hydra
What if we copy the subtree some number of times?

Let me rephrase:

"Say we have a rooted tree with a finite number of nodes and an argument N. After we chop off a leaf A, we consider the following rules:


 * 1) Call the parent of the leaf B. If B is the root, we simply do nothing.
 * 2) If B is not the root, call B's parent C. Take the subtree that contains B and all of its descendants. Copy that subtree N times and attach the new subtrees to C.

GH(m,n) is the largest number of steps to reduce a hydra with length m to the root with argument n."