In this paper we present a new algorithm for merging two linearly ordered sets which requires substantially fewer comparisons than the commonly used tape merge or binary insertion algorithms. Bounds on the difference between the number of comparisons required by this algorithm and the information theory lower bounds are derived. Results from a computer implementation of the new algorithm are given and compared with a similar implementation of the tape merge algorithm.

We provide asymptotics for the range R n of a random walk on the d-dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that n −1 R n converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4 and in the case of a symmetric random walk with exponential moments, we prove that R n grows like n/ log n. We apply our results to asymptotics for the range of branching random walk when the initial size of the population tends to infinity.Keywords. Tree-indexed random walk, range, discrete snake, branching random walk, subadditive ergodic theorem.

We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index α ∈ (1, 2]. Here the harmonic measure refers to the hitting distribution of height n by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation n. For a ball of radius n centered at the root, we prove that, although the size of the boundary is roughly of order n 1 α−1 , most of the harmonic measure is supported on a boundary subset of size approximately equal to n βα , where the constant β α ∈ (0, 1 α−1 ) depends only on the index α. Using an explicit expression of β α , we are able to show the uniform boundedness of (β α , 1 < α ≤ 2). These are generalizations of results in a recent paper of Curien and Le Gall [6]. construct ∆ (α) , one starts with an oriented line segment of length U ∅ , whose origin will be the root of the tree. We call K ∅ the offspring number of the root ∅. Correspondingly, at the other end of the first line segment, we attach the origins of K ∅ oriented line segments with respective lengths U 1 , U 2 , . . . , U K∅ , such that, conditionally given U ∅ and K ∅ , the variables U 1 , U 2 , . . . , U K∅ are independent and uniformly distributed over [0, 1 − U ∅ ]. This finishes the first step of the construction. In the second step, for the first of these K ∅ line segments, we independently sample a new offspring number K 1 distributed as θ α , and attach K 1 new line segments whose lengths are again independent and uniformly distributed over [0, 1 − U ∅ − U 1 ], conditionally on all the random variables appeared before. For the other K ∅ − 1 line segments, we repeat this procedure independently. We continue in this way and after an infinite number of steps we get a random non-compact rooted R-tree, whose completion is the random compact rooted R-tree ∆ (α) . We will call ∆ (α) the reduced stable tree of parameter α. See Section 2.1 for a more precise description. Notice that all the offspring numbers involved in the construction of ∆ (2) are a.s. equal to 2, which correspond to the binary branching mechanism. In contrast, this is no longer the case when 1 < α < 2.We denote by d the intrinsic metric on ∆ (α) . By definition, the boundary ∂∆ (α) consists of all points of ∆ (α) at height 1. As the continuous analogue of simple random walk, we can define Brownian motion on ∆ (α) starting from the root and up to the first hitting time of ∂∆ (α) . It behaves like linear Brownian motion as long as it stays inside a line segment of ∆ (α) . It is reflected at the root of ∆ (α) and when it arrives at a branching point, it chooses each of the adjacent line segments with equal probabilities. We define the (continuous) harmonic measure µ α as the (quenched) distribution of the first hitting point of ∂∆ (α) by Brownian motion.Theorem 2. For every index α ∈ (1, 2], with the same constant β α as in Theorem 1, we have P-a.s. µ α (dx)-a.e., lim r↓0 log µ α (B d (x, r)where B d (x, r) ...