On the distribution of the number of vertices in layers of random trees. (1991)

• Record Type: Journal Article
• Authors: Takács, Lajos
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 4:Number 3(1991)
• Page Start: 175
• Page End: 186
• Publication Date: 1991
• Subjects: random trees -- branching processes -- Bernoulli excursion -- Brownian excursion -- local times -- limit theorems; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S104895339100014X
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : Denote by S n the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set S n, assuming that all the possible n n − 1 choices are equally probable. Define τ n ( m ) as the number of vertices in layer m, that is, the number of vertices at a distance m from the root of the tree. The distance of a vertex from the root is the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τ n ( m ) and their asymptotic behavior in the case where m = [ 2 α n ], 0 < α < ∞ and n → ∞ . In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.