Based on uniform recursive trees, we introduce random trees with the factor of time, which are named Yule recursive trees. The structure and the distance between the vertices in Yule recursive trees are investigated in this paper. For arbitrary time t > 0, we first give the probability that a Yule recursive tree Yt is isomorphic to a given rooted tree γ ; and then prove that the asymptotic dist...

We present the results concerning the embedding of trees into recursive circulants. Recursive circulant G(N,:,/) is a circulant graph with N vertices and jumps of powers of d. We present dilation 1 embedding of Fibonacci trees and full quaternaly trees in G(Z”, 2), and full binary trees and hinomial trees in G(2”,4).

In this work we provide a combinatorial analysis of bucket recursive trees, which have been introduced previously as a natural generalization of the growth model of recursive trees. Our analysis is based on the description of bucket recursive trees as a special instance of so called bucket increasing trees, which is a family of combinatorial objects introduced in this paper. Using this combinat...

We introduce random recursive trees, where deterministically weights are attached to the edges according to the labeling of the trees. We will give a bijection between recursive trees and permutations, which relates the arising edge-weights in recursive trees with inversions of the corresponding permutations. Using this bijection we obtain exact and limiting distribution results for the number ...

We study depth properties of a general class of random recursive trees where each node i attaches to the random node biXic and X0, . . . , Xn is a sequence of i.i.d. random variables taking values in [0, 1). We call such trees scaled attachment random recursive trees (sarrt). We prove that the height Hn of a sarrt is asymptotically given by Hn ∼ αmax logn where αmax is a constant depending only...

ABSTRACT. In this work we provide a combinatorial analysis of bucket recursive trees, which have been introduced previously as a natural generalization of the growth model of recursive trees. Our analysis is based on the description of bucket recursive trees as a special instance of so called bucket increasing trees, which is a family of combinatorial objects introduced in this paper. Using thi...

We study depth properties of a general class of random recursive trees where each node n attaches to the random node !nXn" and X0, . . . , Xn is a sequence of i.i.d. random variables taking values in [0, 1). We call such trees scaled attachment random recursive trees (SARRT). We prove that the height Hn of a SARRT is asymptotically given by Hn ∼ αmax log n where αmax is a constant depending onl...

We study weighted path lengths (depths) and distances for increasing tree families. For those subclasses of increasing tree families, which can be constructed via an insertion process, e.g., recursive trees, plane-oriented recursive trees and binary increasing trees, we can determine the limiting distribution which can be characterized as a generalized Dickman’s infinitely divisible distribution.