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Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
A recurrence related to trees
 Proceedings of the American Mathematical Society
, 1989
"... Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer c ..."
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Cited by 34 (4 self)
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Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer coefficients, which may be called the "tree polynomials." There are n" ~ (n 1)! sequences of edges between vertices (0.1) ux—vx.un_x—vn_x, \<uk<vk<n, that define a free tree on {1,...,«}, because there are n" ~ free trees on n labeled vertices and every such tree has n 1 edges. If we consider each of these n (n — 1)! sequences to be equally likely, the probability that unX and vn_x belong respectively to components of sizes k and n k based on the first « 2 edges is '^oer^r'■•<*< • ■ Knuth and Schönhage [9, §§912] considered treeconstruction algorithms whose analysis depended on the solution of the recurrence (°3) Xn=Cn+ E Pnk(xk+Xnk) 0<k<n for various sequences (cn). The purpose of the present note is to extend the results of [9] and to consider related sequences of functions whose exact and asymptotic values arise in a variety of algorithms. Much of the analysis below, as in [9], depends on properties of the formal power series tt\A \ TV \ \r^n"~[z " 2, 3 3, 8 4, 125 5, (0.4) T(z) = ^ — — = z + z +z +^z + —z +■■ ■, n>l Received by the editors March 18, 1988.
The Minimal Spanning Tree In A Complete Graph And A Functional Limit Theorem For Trees In A Random Graph.
, 1997
"... . The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tr ..."
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Cited by 22 (3 self)
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. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
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Cited by 20 (0 self)
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The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
MSO Zero One Laws on Random Labelled Acyclic Graphs
 Discrete Math
"... Key words: random labelled trees, monadic second order zeroone laws, second order fraisseehrenfeucht games, second moment method PACS: 1 ..."
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Cited by 4 (2 self)
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Key words: random labelled trees, monadic second order zeroone laws, second order fraisseehrenfeucht games, second moment method PACS: 1
Perpendicular dissections of space
"... Abstract. For each pair (Qi, Qj) of reference points and each real number r there is a unique hyperplane h ⊥ QiQj such that d(P, Qi) 2 − d(P, Qj) 2 = r for points P in h. Take n reference points in dspace and for each pair (Qi, Qj) a finite set of real numbers. The corresponding perpendiculars form ..."
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Cited by 2 (1 self)
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Abstract. For each pair (Qi, Qj) of reference points and each real number r there is a unique hyperplane h ⊥ QiQj such that d(P, Qi) 2 − d(P, Qj) 2 = r for points P in h. Take n reference points in dspace and for each pair (Qi, Qj) a finite set of real numbers. The corresponding perpendiculars form an arrangement of hyperplanes. We explore the structure of the semilattice of intersections of the hyperplanes for generic reference points. The main theorem is that there is a real, additive gain graph (this is a graph with an additive real number associated invertibly to each edge) whose set of balanced flats has the same structure as the intersection semilattice. We examine the requirements for genericity, which are related to behavior at infinity but remain mysterious; also, variations in the construction rules for perpendiculars. We investigate several particular arrangements with a view to finding the exact numbers of faces of each dimension. The prototype, the arrangement of all perpendicular bisectors, was studied by Good and Tideman, motivated by a geometric voting theory. Most of our particular examples are suggested by extensions of that theory in which voters exercise finer discrimination. Throughout, we propose many research problems. Postpublication revision 8 May 2002: added reference to Voronoi in §9: “that goes back to the original paper [25a]”. 1.
On the length of a random minimum spanning tree.
, 2013
"... We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→ ∞ E(Ln) = ζ(3) and show that E(Ln) = ζ(3) + c1 c2+o(1) n n4/3 where c1, c2 are expl ..."
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We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→ ∞ E(Ln) = ζ(3) and show that E(Ln) = ζ(3) + c1 c2+o(1) n n4/3 where c1, c2 are explicitly defined constants. 1
AN OPERATOR APPROACH TO THE PRINCIPLE OF INCLUSION AND EXCLUSION
"... ABSTRACT. Using an operator approach we derive SylvesterWhitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the SylvesterWhitworth inclusion and exclusion formula to the resolution of the number of elements in exac ..."
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ABSTRACT. Using an operator approach we derive SylvesterWhitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the SylvesterWhitworth inclusion and exclusion formula to the resolution of the number of elements in exactly mi sets of A's and mi sets of B's respectively. The formula are applied to the complete graph and complete bipartite graph. The enumeration of spanning subgraphs with any preassigned number of disconnected cycles is solved, together with the case where any preassigned number of vertices have degree one.