We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n?2 new un is added to its parent chosen randomly according some rule. In weighted tree, we choose uk among {u1,u2,…,un?1} with probability proportional wk, where (wn)n?1 deterministic sequence that fix beforehand. affine preferential attachment fitnesses, choosing an...

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached parent chosen randomly proportionally its weight. Under some assumptions on the sequence of weights, first order for height such has been recently established one authors. In this paper, we obtain second and third orders in asymptotic expansion weighted trees, und...

The generalized recursive circulant networking can be widely used in the design and implementation of interconnection networks. It consists a series processors, each is connected through bidirectional, point-to-point communication channels to different neighbors. In this work, we apply shortest path routing concept build independent spanning trees on graphs. proposed strategy loosen restricted ...

Increasing trees have been introduced by Bergeron, Flajolet and Salvy [1]. This kind of notion covers several well knows classes of random trees like binary search trees, recursive trees, and plane oriented (or heap ordered) trees. We consider the height of increasing trees and prove for several classes of trees (including the above mentioned ones) that the height satisfies EHn ∼ γ logn (for so...

Abstract We explore the tree limits recently defined by Elek and Tardos. In particular, we find for many classes of random trees. give general theorems three conditional Galton–Watson trees simply generated trees, split generalized (as here), a continuous-time branching process. These results include, example, labelled ordered recursive preferential attachment binary search

By a theorem of Dobrow and Smythe, the depth of the kth node in very simple families of increasing trees (which includes, among others, binary increasing trees, recursive trees and plane ordered recursive trees) follows the same distribution as the number of edges of the form j−(j+1) with j < k. In this short note, we present a simple bijective proof of this fact, which also shows that the resu...

For a martingale (Xn) converging almost surely to a random variable X , the sequence (Xn − X) is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of Régnier’s martingale for the path length in random binary search trees. Grübel and Kabluchko [2014, preprint, arXiv 1410.0469] gave an alte...

Trees are defined as connected graphs without cycles, and their properties are basics of graph theory. A tree on n nodes labeled 1, 2,..., n is a recursive tree if the node labeled 1 is distinguished as the root, and for each 2 ≤ k ≤ n, the labels of the nodes in the unique path from the root to the node labeled k form an increasing sequence [6]. Bucket trees are a generalization of the ordinar...