For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem and the Aldous–Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.

We study several parameters of a random Bienaymé–Galton–Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma ^2$ . Let $f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable $\xi$ . We show that the independence number is in probability asymptotic to $qn$ , where $q$ is the unique solution to $q = f(1-q)$ . One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to $\log n/\log (1/f'(1-q))$ . Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If $p_1 = \mathbb{P}\{\xi =1\}\gt 0$ , then we show that the maximum leaf-height over all nodes in $T_n$ is in probability asymptotic to $\log n/\log (1/p_1)$ . If $p_1 = 0$ and $\kappa$ is the first integer $i\gt 1$ with $\mathbb{P}\{\xi =i\}\gt 0$ , then the leaf-height is in probability asymptotic to $\log _\kappa \log n$ .

We describe a general framework for realistic analysis of sorting algorithms, and we apply it to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort, BubbleSort). Usually the analysis deals with the mean number of key comparisons, but here we view keys as words produced by the same source, which are compared via their symbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n2/4 and BubbleSort to n2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ (n log2n), Θ(n2), Θ (n2 log n). In these three cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely entropy and coincidence) with respect to the algorithm.

We study computational aspects of repulsive Gibbs point processes, which are probabilistic models of interacting particles in a finite-volume region of space. We introduce an approach for reducing a Gibbs point process to the hard-core model, a well-studied discrete spin system. Given an instance of such a point process, our reduction generates a random graph drawn from a natural geometric model. We show that the partition function of a hard-core model on graphs generated by the geometric model concentrates around the partition function of the Gibbs point process. Our reduction allows us to use a broad range of algorithms developed for the hard-core model to sample from the Gibbs point process and approximate its partition function. This is, to the extent of our knowledge, the first approach that deals with pair potentials of unbounded range.