We prove that the number of fringe subtrees, isomorphic to a given tree, in uniformly random trees with given vertex degrees, asymptotically follows a normal distribution. As an application, we establish the same asymptotic normality for random simply generated trees (conditioned Galton-Watson trees). Our approach relies on an extension of Gao and Wormald's (2004) theorem to the multivariate setting.

The k-cut number of rooted graphs was introduced by Cai et al. [Cai and Holmgren, 2019] as a generalization of the classical cutting model by Meir and Moon [Meir and Moon, 1970]. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converge 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 [Janson, 2006].

We consider the model of random trees introduced by Devroye [Devroye, 1999], the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. The approach we develop may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we have also studied the case of complete d-regular trees.