Abstract
AbstractWe explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.
Highlights
Elek and Tardos [20] have recently introduced a theory of tree limits, in analogy with the theory of graph limits [39] and other similar limits of various combinatorial objects
Their idea is to regard a tree as a metric space with a probability measure; the metric is the usual graph distance, suitably rescaled, and the probability measure is the uniform measure on the vertices
We find tree limits in all these cases, as the size n → ∞; in some cases with convergence in distribution to a random tree limit, and in others with convergence in probability to a fixed tree limit
Summary
Elek and Tardos [20] have recently introduced a theory of tree limits, in analogy with the theory of graph limits [39] and other similar limits of various combinatorial objects (e.g. hypergraphs, permutations, . . . ). For a class of condition√ed Galton–Watson trees including many standard classes of random trees with height of order n (Section 9), the well-known limit theorem by Aldous [7] gives convergence to a random real tree known as the the Brownian continuum random tree; this tree can be regarded as a (random) long dendron, and Aldous’s result holds in the present sense too. Many standard classes of random trees with height of order log n are covered by the general results in Sections 12–14 and have tree limits of a quite different type; these limits are long dendrons of a very simple type (but distinct from real trees), which is equivalent to the fact that in these trees, almost all pairs of vertices have almost the same distance. There are cases (see for example Section 10) where the trees are such that there is a strong relation between the tree limits and local limits
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