A tree is called k-decomposable if it has a spanning forest whose components are all of size k. In this paper, we study the number of k-decomposable trees in families of increasing trees, i.e. labeled trees in which the unique path from the root to an arbitrary vertex forms an increasing sequence. Functional equations for the corresponding counting series are provided, yielding asymptotic or ev...

In a recent paper, Shah and Zaman proposed the rumor center as an effective rumor source estimator for rumor spreading on random graphs. They proved for a very general random tree model that the detection probability remains positive as the number of nodes to which the rumor has spread tends to infinity. Moreover, they derived explicit asymptotic formulas for the detection probability of random...

A periodic tree Tn consists of full n-level copies of a finite tree T . The tree Tn is labeled by random bits. The root label is chosen randomly, and the probability of two adjacent vertices to have the same label is 1− . This model simulates noisy propagation of a bit from the root, and has significance both in communication theory and in biology. Our aim is to find an algorithm which decides ...

Simple families of increasing trees have been introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees...

Stochastic trees are semi-Markov processes represented using tree diagrams. Such trees have been found useful for prescriptive modeling of temporal medical treatment choice. We consider utility functions over stochastic trees which permit recursive evaluation in a graphically intuitive manner analogous to decision tree rollback. Such rollback is computationally intractable unless a low-dimensio...

Let Tn be a random recursive tree with vertex set [n] := {1, . . . , n} and edges directed towards the root. Let degn(i) denote the number of children of vertex i ∈ [n] of Tn. Devroye and Lu [1] showed that the maximum degree ∆n of Tn satisfies ∆n/ log2 n→ 1 a.s. Here we study the distribution of the maximum degree and of the number of vertices with near-maximum degree. For any d ∈ Z, let X d =...

We present a general construction to reduce the communication cost of recent tree-based ORAMs. Contrary to trees with constant height and path lengths, our new construction r-ORAM provides varying, shorter path lengths. Accessing an element in the ORAM tree will have different communication cost depending on the location of the element. The main idea behind r-ORAM is a recursive ORAM tree struc...

Khoussainov, B. Recursive “nary algebras and trees, Annals of Pure and Applied Logic 67 (1994) 213-268. A unary algebra is an algebraic system d = (A,f,, ,L). wheref,, . ,fm are unary operations on A and n EW. In the paper we develop the theory ofeffective “nary algebras. We investigate well-known questions of constructive (recursive) model theory with respect to the class of unary algebras. In...

We consider the number of random cuts that are necessary to isolate the node with label λ, 1 ≤ λ ≤ n, in a random recursive tree of size n. At each stage of the edge-removal procedure considered an edge is chosen at random from the tree and cut, separating the tree into two subtrees. The procedure is then continued with the subtree containing the specified label λ, whereas the other subtree is ...

While tree contraction algorithms play an important role in efficient tree computation in parallel, it is difficult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving efficient tree contraction algorithms from recursive functions on trees. We identify a general recursive form that can be parallelize...