Death of a monster

The Hydra was a monster with many heads, living in a marsh of Lerna. Hercules fought it in a formidable struggle, chopping off its heads with a club. However, as Hercules chopped a head, two new heads grew forth from the monster 's body. According to the official records Hercules won. What the Greek did not reveal is that , no mat ter in which order Hercules had decided to attack the heads, he would have annihilated the Hydra. Although less charming and botanically incorrect, we treat the Hydra as a rooted tree and the heads as leaves (figure 1). For a leaf z the predecessor y is the neck, and the predecessor's predecessor z is the trunk. When z is cut off two new subtrees, rooted in the new necks u and v, grow from the trunk z. These subtrees are identical to the one rooted in y (deprived of x). The first relevant fact is that , independently of the order in which the leaves are cut, the tree eventually reduces to its root after a f~nlte number of cuts, and then disappears with the next cut. Before we prove the.q, the reader may practice a little fighting the Hydra of figure I. Even if the battle is not brought to its end, as it may take a huge number of steps, the following facts may help understanding the nature of the process: