FIGURE 7 - uploaded by Matas Šileikis
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The different types used to study the number of leaves in a quaternary search tree.
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We study fringe subtrees of random m-ary search trees and of preferential attachment trees, by putting them in the context of generalised Polya urns. In particular we show that for the random m-ary search trees with m <= 26 and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree...
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Citations
... This means that log | Aut T | is an additive functional of the tree (see [5,Sect. 3.2] or any of the references below) with toll function f (T ) = k i=1 log(m i !). Limit theorems for additive functionals have been proven for various classes of random trees under different conditions, see [6,7,10,19,20,22]. In the case of Galton-Watson trees, we will specifically make use of a general result on almost local additive functionals due to Ralaivaosaona, Šileikis and the second author [20], which is in turn based on earlier work by Janson [10]. ...
We study the size of the automorphism group of two different types of random trees: Galton–Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for the mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton–Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.
... When = 1, the tree n is called a random plane-oriented recursive tree which was introduced by Szymański [32]. The more general linear preferential attachment tree ,n coincides with a special case of the preferential attachment model studied by Barabási and Albert [2], but has also been studied in several other contexts (see e.g., [7,17,31]). When = − 1 for a positive integer , the tree −1∕ ,n is a model of random d-ary trees, and corresponds to a random binary search tree when = 2. ...
... The simplest example of a fringe subtree in is a vertex with no descendants (a leaf of ). Normal limit laws for the number of leaves in preferential attachment trees are already well known (see [17,19,24,29]). ...
... Limiting joint distributions for the number of fringe subtrees (without colors) have already been studied [17]. Let T 1 , … , T m be a sequence of finite trees of sizes k 1 , … , k m with colorings 1 , … , m . ...
We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability p$$ p $$ and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.
... where the sum is over root branches up to isomorphism. In this case the toll function is f (T ) = log(m i !). Limit theorems for additive functionals have been proven for various classes of random trees under different conditions, see [6,7,10,[20][21][22]. In the case of Galton-Watson trees, we will specifically make use of a general result on almost local additive functionals due to Ralaivaosaona,Šileikis and the second author [21], which is in turn based on earlier work by Janson [10]. ...
We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted P\'olya trees. We also show how to extend the results to some classes of unrooted trees.
... Once again, we make use of results on additive functionals. For additive functionals of increasing trees with finite support, i.e., for functionals for which there exists a constant K such that f (t) = 0 whenever |t| > K , a central limit was proven in [28] and [42] (the latter even contains a slightly more general result). Those results do not directly apply to the additive functionals that we are considering here. ...
... Proof The first statement follows directly from Lemma 3, since fringe subtrees are, conditioned on their size, again random trees following the same probabilistic model as the whole tree. For functionals with finite support, where f (T ) = 0 for all but finitely many trees T , convergence in probability follows from the central limit theorems in [28] and [42]. For the more general case, we approximate the additive functional F with a truncated version F m based on the toll function ...
A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in random trees under a generalized notion of distinctness, which allows for many different interpretations of what “distinct” trees are. The random tree models considered are simply generated trees and families of increasing trees (recursive trees, d -ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what ‘distinct’ means) in random trees with n vertices is $$n/\sqrt{\log n}$$ n / log n for simply generated trees and $$n/\log n$$ n / log n for increasing trees.
... When α = 1, the tree T n is called a random plane-oriented recursive tree which was introduced by Szymański [31]. The more general linear preferential attachment tree T α,n coincides with a special case of the preferential attachment model studied by Barabási and Albert [2], but has also been studied in several other contexts (see for example [30,7,17]). When α = − 1 d for a positive integer d, the tree T −1/d,n is a model of random d-ary trees, and corresponds to a random binary search tree when d = 2. ...
... The simplest example of a fringe subtree in T is a vertex with no descendents (a leaf of T ). Normal limit laws for the number of leaves in preferential attachment trees are already well known (see [28,23,19,17]). ...
... Limiting joint distributions for the number of fringe subtrees (without colours) have already been studied [17]. Let T 1 , . . . ...
In this work we consider random two-colourings of random linear preferential attachment trees, which includes random recursive trees, random plane-oriented recursive trees, random binary search trees, and a class of random $d$-ary trees. The random colouring is defined by assigning the root of the tree the colour red or blue with equal probability, and all other vertices are assigned the colour of their parent with probability $p$ and the other colour otherwise. These colourings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of P\'olya urns, we prove limiting distributions, after proper rescalings, for the number of vertices of each colour, the number of monochromatic subtrees of each colour, as well as the number of leaves and fringe subtrees with two-colourings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root. The description of the limiting distributions extends previous work on bond percolation in random preferential attachment trees.
... Once again, we make use of results on additive functionals. For additive functionals of increasing trees with finite support, i.e., for functionals for which there exists a constant K such that f (t) = 0 whenever |t| > K, a central limit was proven in [26] and [39] (the latter even contains a slightly more general result). Those results do not directly apply to the additive functionals that we are considering here. ...
... The first statement follows directly from Lemma 3, since fringe subtrees are, conditioned on their size, again random trees following the same probabilistic model as the whole tree. For functionals with finite support, where f (T ) = 0 for all but finitely many trees T , convergence in probability follows from the central limit theorems in [26] and [39]. For the more general case, we approximate the additive functional F with a truncated version F m based on the toll function ...
A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in two types of random trees: simply generated trees and families of increasing trees (recursive trees, $d$-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what `distinct' means) in random trees with $n$ vertices is $n/\sqrt{\log n}$ for simply generated trees and $n/\log n$ for increasing trees.
... Note that even though the sum is formally infinite, for any T only a finite number of F S (T ) are nonzero. Functionals of the form F S are known to be asymptotically normally distributed in various classes of trees, notably simply generated trees/Galton-Watson trees [9,20], which are the topic of this paper, and several other models of random trees [5,6,16]. In view of this and several other important examples of additive functionals that satisfy a central limit theorem, general schemes have been devised that yield a central limit theorem under different technical assumptions. ...
An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Svante Janson recently proved a central limit theorem for additive functionals of conditioned Galton–Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are “almost local” in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph-theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction procedure that was studied by Hackl, Heuberger, Kropf, and Prodinger.
... Both models are examples of preferential attachment trees, where the choice of v is made proportionally to χ deg(v) + ρ for real parameters χ and ρ (notice that a preferential attachment tree is a random recursive tree when χ = 0 and is a plane-oriented random recursive tree when ρ = 0). Pólya urns were used to prove multivariate normal limit laws for the degree distributions in all of these random tree models [4,6,9,10]. ...
... is the number of children of v. But we can simply let ρ = ρ ′ − χ to get the same model, and replace w k with w ′ k−1 = χ(k − 1) + ρ ′ so that (10) resembles more the statements of the previous results [6,4,9,10]. The only vertex where this does not translate is the root (or master hook) of the network, since deg(H) = deg + (H) in this case, but see Remarks 2.2 and 3.4 below for why this doesn't affect the limiting distribution. ...
... The Pólya urn described above has infinitely many types, and so Theorem 2.1 does not apply. Therefore, we would like to instead use an urn with finitely many types in the same manner as is done in [4] and [6]. The urn is replaced with the following Pólya urn: let d be a positive integer corresponding to the largest (out)degree we wish to study in this instance of the model. ...
We consider two types of random networks grown in blocks. Hooking networks are grown from a set of graphs as blocks, each with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network and a copy of one of the blocks is attached by fusing its hook with the latch. Bipolar networks are grown from a set of directed graphs as blocks, each with a single source and a single sink. At each step in the growth of the network, an arc is chosen and is replaced with a copy of one of the blocks. Using P\'olya urns, we prove normal limit laws for the degree distributions of both networks. We extend previous results by allowing for more than one block in the growth of the networks and by studying arbitrarily large degrees.
... We use basic assumptions on the Pólya urn, these are (A1)-(A7) gathered in [6] ((A1)-(A6) are also explicitly stated in [8]). A ball of type i is said to be dominating if with positive probability, every other ball of type j can be found at some time in an urn starting with a single ball of type i. ...
... The Pólya urn described above has infinitely many types, and so Theorem 2.1 does not apply. We would like to instead use an urn with finitely many types in a similar manner as was done in [6] and [9]. The urn is replaced with the following Pólya urn: let d be a positive integer corresponding to the largest degree we wish to study in this instance of the model. ...
... Functionals of the form F S are known to be asymptotically normally distributed in different classes of trees, notably simply generated trees/Galton-Watson trees [8,18], which will also be the topic of this paper, and classes of increasing trees [5,14]. In view of this and several other important examples of additive functionals that satisfy a central limit theorem, general schemes have been devised that yield a central limit theorem under different technical assumptions. ...
... Previous results [5,8,14,18], while giving rather general conditions on the toll function that imply normality, are unfortunately still insufficient to cover all possible examples one might be interested in. This paper is essentially an extension of Janson's work [8] on local functionals. ...
An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are "almost local" in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger.
