Results 1  10
of
18
Random cutting and records in deterministic and random trees
 ALG
, 2006
"... We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Wat ..."
Abstract

Cited by 48 (9 self)
 Add to MetaCart
(Show Context)
We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random, and the case when we condition on the tree. The proofs are based on Aldous’ theory of the continuum random tree.
Random recursive trees and the BolthausenSznitman coalescent
 Electron. J. Probab
, 2005
"... We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the ..."
Abstract

Cited by 34 (2 self)
 Add to MetaCart
(Show Context)
We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n → ∞, and obtain a scaling law for the sizes of these blocks. We also consider the discretetime Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the timereversal of this Markov chain have limits as n → ∞. These results can be interpreted as describing a “postgelation ” phase of the BolthausenSznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. 1
Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent
, 2007
"... ..."
Asymptotic results about the total branch length of the BolthausenSznitman coalescent
"... We study the total branch length Ln of the BolthausenSznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random v ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
We study the total branch length Ln of the BolthausenSznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the BolthausenSznitman coalescent with mutation rate r> 0. Moreover, the results show that, for the BolthausenSznitman coalescent, the total branch length Ln is closely related to Xn, the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.
MULTIPLE ISOLATION OF NODES IN RECURSIVE TREES
"... ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a size ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a sizen random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled 1, 2..., ℓ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled n + 1 − ℓ, n + 2 − ℓ,... n (thus nodes located at the fringe of the tree), and (iii) isolating ℓ nodes in the tree, which are selected at random before starting the edgeremoval procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behaviour of the number of cuts to isolate all selected nodes, for ℓ fixed and n → ∞. 1.
A weakly 1stable limiting distribution for the number of random records and cuttings in split trees
"... We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the dis ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1stable. This work is a generalization of our earlier results for the random binary search tree in [10], which is one specific case of split trees. Other important examples of split trees include mary search trees, quadtrees, medians of (2k+ 1)trees, simplex trees, tries and digital search trees.
Cutting down trees with a Markov chainsaw
, 2011
"... We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finitevariance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, nonasymptotic distributio ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finitevariance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton–Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge. 1