# The $k$-cut model in deterministic and random trees

## TL;DR

This paper studies the k-cut number in various rooted trees, showing that after rescaling, it converges in distribution or probability, revealing universal behaviors across different tree models.

## Contribution

It extends existing results by proving convergence of moments for the k-cut number in conditioned Galton-Watson trees and other tree types, regardless of offspring distribution.

## Key findings

- Moments of k-cut number converge after rescaling in conditioned Galton-Watson trees.
- k-cut number converges to a constant in various logarithmic height trees.
- Results hold for both deterministic and random tree models.

## Abstract

The $k$-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the k-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.