Fragmentation of random trees

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected existing node. As nodes are removed from the tree, one at a time, the tree dissolves into an ensemble of separate trees, namely, a forest. We study statistical properties of trees and nodes in this heterogeneous forest, and find that the fraction of remaining nodes m characterizes the system in the limit N→∞. We obtain analytically the size density ϕs of trees of size s. The size density has power-law tail ϕs∼s−α with exponent α=1+(1/m). Therefore, the tail becomes steeper as further nodes are removed, and the fragmentation process is unusual in that exponent α increases continuously with time. We also extend our analysis to the case where nodes are added as well as removed, and obtain the asymptotic size density for growing trees. (paper)