Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when the underlying tree is a size-conditioned Galton-Watson tree and calculate its probability of being correct.
We study several parameters of a random Bienaymé–Galton–Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma ^2$ . Let $f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable $\xi$ . We show that the independence number is in probability asymptotic to $qn$ , where $q$ is the unique solution to $q = f(1-q)$ . One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to $\log n/\log (1/f'(1-q))$ . Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If $p_1 = \mathbb{P}\{\xi =1\}\gt 0$ , then we show that the maximum leaf-height over all nodes in $T_n$ is in probability asymptotic to $\log n/\log (1/p_1)$ . If $p_1 = 0$ and $\kappa$ is the first integer $i\gt 1$ with $\mathbb{P}\{\xi =i\}\gt 0$ , then the leaf-height is in probability asymptotic to $\log _\kappa \log n$ .
Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when the underlying tree is a size-conditioned Galton-Watson tree and calculate its probability of being correct.
In this paper we investigate properties of the lattice Ln of subsets of [n] = {1, . . . , n} that are arithmetic progressions, under the inclusion order. For n ≥ 4, this poset is not graded and thus not semimodular. We start by deriving properties of the number p nk of arithmetic progressions of length k in [n]. Next, we look at the set of chains in L ′ n = Ln \ {∅, [n]} and study the order complex ∆n of L ′ n . Third, we determine the set of coatoms in Ln to give a general formula for the value of µn evaluated at an arbitrary interval of Ln. In each of these three sections, we give an independent proof of the fact that for n ≥ 2, µn(Ln) = µ(n − 1), where µn is the Möbius function of Ln and µ is the classical (number-theoretic) Möbius function. We conclude by computing the homology groups of ∆n, providing yet another explanation for the formula of the Möbius function of Ln.
We study several parameters of a random Bienaymé-Galton-Watson tree Tn of size n defined in terms of an offspring distribution ξ with mean 1 and nonzero finite variance σ 2 . Let f (s) = E{s ξ } be the generating function of the random variable ξ. We show that the independence number is in probability asymptotic to qn, where q is the unique solution to q = f (1 − q). One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to log n/ log (1/f ′ (1 − q)).Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If p1 = P{ξ = 1} > 0, then we show that the maximum leaf-height over all nodes in Tn is in probability asymptotic to log n/ log(1/p1). If p1 = 0 and κ is the first integer i > 1 with P{ξ = i} > 0, then the leaf-height is in probability asymptotic to log κ log n.
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