ANALYSIS OF SOME TREE STATISTICS

Transcription

1 ANALYSIS OF SOME TREE STATISTICS ALOIS PANHOLZER Abstract. In this survey we present results and the used proof techniques concerning the analysis of several parameters in frequently used tree families. In particular we consider first the two tree statistics ancestor-tree size resp. Steiner-distance of p nodes, that extend in a natural way the parameters depth of a node resp. distance between two nodes. Under the assumption, that p nodes are selected at random, we can show limiting distribution results for a lot of tree models. The so called model of random search trees allows a reformulation of that result to parameters in the Multiple Quickselect algorithm, which is also given. Another analysed tree statistic is the number of steps, that are necessary to climb a random tree of size n. An equivalent formulation of that parameter is the size of incomplete resp. one-sided versions of the considered trees, which has been studied quite recently by several authors. Here we consider this parameter for different tree models and also show limiting distribution results. The last here considered tree statistic deals with the number of random cuts, that are necessary to destroy a tree of size n. Again, we can show for certain tree models limiting distribution results.. Introduction.. Considered tree statistics. We will give in this paper results concerning the following tree parameters.... Ancestor-tree size and Steiner-distance. These are two parameters which extend the quantities depth of a random node and distance between two random nodes for rooted trees. The depth of a node v is defined here as the number of nodes lying on the unique path from the root to v. The natural extension size of the ancestor-tree of p chosen nodes v,..., v p measures the size of the tree spanned by the root and v,..., v p and therefore counts the number of nodes that are lying on at least one direct path from the root to v i for i p. The node distance between v and v 2 is defined here as the number of nodes lying on the direct path from v to v 2. A natural extension considered here is the spanning subtree size of p chosen nodes v,..., v p and thus counts the number of nodes that lie on at least one direct path from v i to v j for i j p. In the literature, this parameter is also known as the Steiner distance between these p nodes. See Figure for a comparison of both parameters considered here: ancestor-tree size and Steiner-distance...2. Climbing trees. We are considering the following procedure to climb rooted trees. We start with a tree T of size n rooted at node r 0, where the size measures as usual the number of nodes of T. If n 2, then we choose one of the edges in the tree that are incident with the root r 0 say r 0, r and proceed along it to node r. Then we iterate this procedure with the subtree of T rooted at node r : if the size of this subtree is greater than r is not Date: May 9, 2004.

2 2 A. PANHOLZER Figure. A rooted tree with the two parameters under consideration. Ancestor-tree size of the 3 marked nodes is 6. Steiner-distance of the 3 marked nodes is 5. an endnode of T, we choose one of the edges incident with r say r, r 2 and proceed to node r 2, and so on. After m n steps = the number of used edges, we will reach an endnode of T and we stop. We are here interested in the parameter number of steps to climb the tree and thus to reach an endnode. An example for climbing a rooted tree is given in Figure 2. r 4 r 3 r 2 r r 0 Figure 2. Climbing in 4 steps a rooted tree of size Cutting down trees. In this paper, we are considering the following cutting down procedure to destroy trees. We start with a tree T of size n. If n 2, then we choose one of the n edges in the tree and remove this edge from T. After removing this edge, the original tree T falls into two subtrees, where one of them, which we will denote by T, contains the root and has size k with k n. If k 2, this edge-removing procedure will be iterated with T : we choose one of the k edges of T and remove it, take the subtree containing the root, and so on. After m n steps we will obtain a tree consisting only of the root itself and we stop. We are here interested in the parameter number of steps to cut down the tree and thus to isolate the root. An example for cutting down a tree is given in Figure 3.

3 ANALYSIS OF SOME TREE STATISTICS 3 Figure 3. Destroying in 5 steps a tree of size, where the dotted lines mark the edges, that will be removed in the next step..2. Used probability models and analysed random variables. For all considered tree statistics and all tree families as defined in Subsection.3., we will always assume the random tree model as the model of randomness. This means, that either for unweighted trees every tree of size n in the considered tree family is selected with equal probability, or when a weight is associated to the trees in the considered tree family, every tree of size n is selected with probability proportional to its weight if there exists a tree of size n with positive weight, otherwise the probability is zero..2.. Ancestor-tree size and Steiner-distance. Here we additionally assume that all n p possibilities of selecting p nodes in a tree of size n are equally likely. Then throughout this paper we will for a given tree family always denote by X n,p the random variable counting the size of the ancestor tree of p randomly chosen nodes in a randomly chosen tree of size n and by Y n,p the random variable counting the Steiner distance of p randomly chosen nodes in a randomly chosen tree of size n. For the so called generalized M-ary search trees we have two additional parameters M and t, thus the according random variables are denoted by X M,t;n,p and Y M,t;n,p. The connections between random search trees and variants of the Quicksort algorithm see e. g. [3, 4] leads to another interpretation of X M,t;n,p : assuming the random permutation model for the input data, this quantity also counts the number of passes that are required in the generalized Multiple Quickselect algorithm see [33] for the basic algorithm and [32] for generalizations to find a random p-order statistic in a data file of length n Number of steps to climb a tree. Here we additionally assume for the climbing procedure described in Subsection..2, that for every node v reached in a tree T, the next edge is chosen at random from the edges incident with v. Then throughout this paper we will for a given tree family always denote by W n the random variable, which counts the number of steps that are used to climb a randomly chosen tree equivalently which counts the number of edges lying on the path from the root to the reached endnode of size n with this randomized climbing procedure.

4 4 A. PANHOLZER.2.3. Number of steps to cut down a tree. Here we additionally assume for the cutting down procedure described in Subsection..3, that the removed edges are at each stage chosen at random from the remaining tree containing the root. Then throughout this paper we will for a given tree family always denote by Z n the random variable, which counts the number of edges that will be removed from a randomly chosen tree of size n with this randomized cutting down procedure until the root is isolated or equivalently which counts the number of steps that are done until the tree is destoyed or cut down..3. Considered tree families. Next, we will describe all tree families which are analyzed here..3.. Simply generated trees. Simply generated trees were introduced in [23] and they include several important tree families, e. g. binary trees, unordered labelled trees = Cayley trees and planted plane trees = ordered trees. Moreover, they are strongly related to Galton- Watson branching processes, since it is well known see [], that random simply generated trees are essentially the same as conditioned Galton-Watson trees, obtained as the family tree of a Galton-Watson process conditioned on the given total size. A class T of simply generated trees can be defined in the following way. A sequence of non-negative real numbers ϕ k k 0 with ϕ 0 > 0 is used to define the weight wt of any ordered tree T by wt := v ϕ dv, where v ranges over all vertices of T and dv is the out-degree the number of children of v. The family T consists then of all trees T with wt 0 together with its weights wt. It follows further, that the generating function T z = n 0 T nz n of the quantity T n := T =n wt, where T denotes the size of the tree T, satisfies the functional equation T z = zϕ T z, where the degree generating function ϕt is given as ϕt = k 0 ϕ kt k. The asymptotic behaviour of T z as solution of is discussed in detail in [9] Recursive Trees. A rooted labelled non-plane tree T of size n with labels, 2,..., n is a recursive tree, if the root is labelled with, and for each node v holds, that the labels of the vertices on the unique path from the root to v form an increasing sequence. One obtains, that the exponential generating function T z = n T n zn n! of the number T n of different recursive trees of size n satisfies the differential equation Of course this gives T z = log results on recursive trees see [8]. T z = e T z, T 0 = 0. 2 z and T n = n!. For a survey of applications and.3.3. Heap ordered trees. A rooted labelled plane tree T of size n with labels, 2,..., n is a heap ordered tree also called plane recursive tree or plane increasing tree, if the root is labelled with, and for each node v holds, that the labels of the vertices on the unique path from the root to v form an increasing sequence. Here by introducing the exponential generating function T z = n T n zn n! of the number T n of different heap ordered trees of size n one obtains the differential equation T z =, T 0 = 0. 3 T z

5 ANALYSIS OF SOME TREE STATISTICS 5 One gets the solution T z = 2z and further T n = n n! k= 2k = For a survey of applications and results on heap ordered trees see also [8]. n. 2 n 2n Increasing trees. Increasing trees are labelled trees, where the nodes of a tree T of size n are labelled by distinct integers of the set {,..., n} in such a way, that each sequence of labels along any branch starting at the root is increasing. As the underlying tree model, we use the previous defined simply generated trees, but additionally, they are equipped with increasing labellings. Thus we call them simple families of increasing trees. A class T of a simple family of increasing trees can be defined in the following way. A sequence of non-negative numbers ϕ k k 0 with ϕ 0 > 0 is used to define the weight wt of any ordered tree T by wt := v ϕ dv, where v ranges over all vertices of T and dv is the out-degree of v. Further, LT denotes the number of different increasing labellings of the tree T. Then the family T consists of all trees T together with their weights wt and the various increasing labellings LT. For a given degree sequence ϕ k k 0, we define the weights T n := T =n wt LT. It follows further, that the exponential generating function T z = n 0 T n zn n! of T n satisfies the autonomous first order differential equation T z = ϕ T z, T 0 = 0, 4 where the degree generating function ϕt is given as ϕt = k 0 ϕ kt k. The previous defined recursive trees and heap ordered trees are members of the here introduced family of increasing trees: they are obtained with degree generating functions ϕt = e t resp. ϕt = t. Another important subclass of the family of increasing trees is the family of polynomial increasing trees. Here the degree generating function ϕt = d k=0 ϕ kt k is a polynomial and thus it exists an integer d, such that the out-degrees of the nodes in every tree are bounded by at most d. The instance ϕt = + t 2 which are called binary increasing trees is of particular interest, since it is well known, that this family is isomorphic to the family of ordinary binary seaach trees as defined in Subsection.3.5. The asymptotic behaviour of T z as solution of 4 if ϕt is a polynomial is discussed in detail in [2] M-ary search trees. Binary search trees are a fundamental data structure which are permanently used in computer science. Generalizations of them are M-ary search trees and fringe balanced M-ary search trees. In M-ary search trees, each node can have up to M keys. The ordinary M-ary search tree T constructed from a given set {K,..., K n } of keys is then obtained in the following way cf. [3]: if n < M then T consists only of a single node, where the n keys are stored in increasing order; if n M then the first M keys are stored in the root of T in increasing order and the remaining n M + keys are stored in a recursive way in the subtrees T,..., T M, that are themselves M-ary search trees, and where all keys with values between K i and K i, for 2 i M, are going to T i, and moreover T 0 will keep all keys smaller than K and T M will keep all keys larger than K M. One can make M-ary search trees more balanced, when selecting first at random Mt+ elements from which the M equi-spaced ranks the t+-st,..., M t+-st smallest are chosen and stored in the root. For ordinary M-ary search trees we have of course t = 0

6 6 A. PANHOLZER in particular ordinary binary search trees are obtained for M = 2 and t = 0. The obtained M-ary search trees are called generalized M-ary search trees or fringe balanced M-ary search trees cf. [4] Unordered unlabelled rooted trees. These trees sometimes called Polya trees can be defined recursively as a root followed by a possibly empty set of rooted trees; thus a subtree structure and all its permutations are just count once. If T n denotes the number of unordered unlabelled rooted trees of size n and T z := n T nz n its ordinary generating function, then Polya [36] showed the functional equation T z = z k 0 ZS k ; T z, T z 2,..., T z k = z k 0 Z k {T z}, 5 where ZS k ; x,..., x k denotes the cycle-index of the symmetric group S k of degree k and where for arbitrary functions fz the following abbreviation is used: Z k {fz} := Z S k ; fz, fz 2,..., fz k. The functional equation 5 can be written as and was studied first by Otter in [26]. T z = z exp k T z k, 6 k.3.7. Non-crossing trees. A non-crossing tree is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, whose edges are straight line segments that do not cross. We consider the vertices labelled clockwise from to n, where the root of the tree is vertex. Fig. 4 gives an example of a non-crossing tree Figure 4. A non-crossing tree of size 2. It is well known, that the number T n of different non-crossing trees of size n is equal to the number of ternary trees of size n. Therefore it holds T n = 3n 3 2n n. 7 In the combinatorial analysis of non-crossing trees via generating functions see [8] one uses T z = n T nz n and Bz = n B nz n, where the auxiliary quantities B n are given by B n = 3n 2 n n.

7 ANALYSIS OF SOME TREE STATISTICS 7 2. Results for the considered tree statistics In the following we state results for the random variables X n,p, Y n,p, W n and Z n, where d we use the following abbreviations: means convergence in distribution, Φx denotes the distribution function of a standard N 0, normally distributed random variable, H n := n k= k and H2 n := n k= denote first and second order harmonic numbers and γ is the k 2 Euler constant. 2.. Results concerning the Ancestor-tree size and the Steiner-distance. The distribution of the depth of a random node and thus X n, has been studied for a lot of tree families. We mention here the works of Mahmoud and Pittel [7] for M-ary search trees, Devroye [5] for recursive trees, Mahmoud for heap ordered trees [4] and Meir and Moon [23] for simply generated trees. The disbribution of the distance between two random nodes and thus Y n,2 has been studied for binary search trees by Mahmoud and Neininger [6] and for recursive trees by Dobrow [6]. Quite recently obtained results for X n,p and Y n,p by the author are given below Limiting distribution for simply generated trees. Theorem. [3] Let ϕt = k 0 ϕ kt k with ϕ 0 > 0 and ϕ k 0 for k > 0, where we suppose that ϕt has a positive radius of convergence R > 0 and assume, that there exists a minimal positive solution τ < R of the equation tϕ t = ϕt. Furthermore we denote ρ = τ ϕτ and define d := gcd{k : ϕ k > 0}. Then for simply generated trees with degree generating function ϕt, the random variable X n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a randomly selected tree of size n and the random variable Y n,p, which counts the Steiner-distance of p randomly chosen nodes in a randomly selected tree of size n, converge in distribution for fixed p resp. p 2 and n provided that n mod d to generalized Gamma distributed random variables: X n,p n d X p, Y n,p n d Y p, where X p = Y p+ and Y p is a random variable with density function f p x = x 2p 3 2 ρτϕ τ p e ρτϕ τ x 2 2, for x 0, and f p x = 0 otherwise. p 2! 2 h If ga, h, A; x = ΓaA x A ah e x A h for x > 0 denotes the density function of the generalized Gamma distribution we get thus that X p resp. Y p have densitiy functions g p, 2, 2 ρτϕ τ ; x resp. g p, 2, 2 ρτϕ τ. ; x Limiting distribution for recursive trees. Theorem 2. [3] The distribution of the random variable X n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a random recursive tree of size n and the distribution of the random variable Y n,p, which counts the Steiner-distance of p randomly

8 8 A. PANHOLZER chosen nodes in a random recursive tree of size n, are for fixed p resp. p 2 and n asymptotically Gaussian, where the rate of convergence is of order O log n : { } Xn,p p log n P = Φx + O, p log n log n { } Yn,p p log n P = Φx + O. p log n log n Limiting distribution for heap ordered trees. Theorem 3. [25] The distribution of the random variable X n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a random heap ordered tree of size n and the distribution of the random variable Y n,p, which counts the Steiner-distance of p randomly chosen nodes in a random heap ordered tree of size n, are for fixed p resp. p 2 and n asymptotically Gaussian, where the rate of convergence is of order O log n : { Xn,p p 2 P log n } < x p 2 log n = Φx + O, log n { Yn,p p 2 P log n p 2 log n < x } = Φx + O. log n Limiting distribution for polynomial increasing trees. Theorem 4. [35] Let ϕt = ϕ 0 + ϕ t + + ϕ d t d with ϕ 0 > 0, ϕ d > 0 and ϕ k 0 for 0 < k < d be a polynomial with degree d 2. Then for increasing trees with degree generating function ϕt, the distribution of the random variable X n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a randomly selected tree of size n and the distribution of the random variable Y n,p, which counts the Steiner-distance of p randomly chosen nodes in a random ly selected tree of size n, are for fixed p resp. p 2 and n asymptotically Gaussian, where the rate of convergence is of order O log n : X n,p pd d P log n < x pd d log n = Φx + O, log n P Y n,p pd d log n pd d log n < x = Φx + O. log n Limiting distribution for generalized M-ary search trees. Theorem 5. [32, 34] The distribution of the random variable X M,t;n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a random generalized M-ary search tree of size n and also counts the number of passes in the generalized Multiple Quickselect algorithm to find a random p-order statistic in a random data array of size n and the distribution of the random variable Y M,t;n,p, which counts the Steiner-distance of p randomly chosen nodes

9 ANALYSIS OF SOME TREE STATISTICS 9 in a random generalized M-ary search tree of size n, are for fixed p resp. p 2 and n asymptotically Gaussian, where the rate of convergence is of order O log n : X M,t;n,p H Mt+ H t+ p log n P < x = Φx + O, H 2 Mt+ H2 t+ log n H Mt+ H t+ p log n 3 P Y M,t;n,p H Mt+ H t+ p log n H 2 Mt+ H2 t+ H Mt+ H t+ 3 p log n Limiting distribution for non-crossing trees. < x = Φx + O. log n Theorem 6. [27] The random variable X n,p, which counts the size of the ancestor-tree of p randomly chosen nodes in a random non-crossing tree of size n and the random variable Y n,p, which counts the Steiner-distance of p randomly chosen nodes in a random non-crossing tree of size n, converge in distribution for fixed p resp. p 2 and n to generalized Gamma distributed random variables: X n,p d X p, n Y n,p n d Y p, where X p = Y p+ and Y p is a random variable with density function f p x = 2x2p 3 3 p e 3 4 x2, for x 0, and f p x = 0 otherwise. p 2! Results concerning the climbing procedure. This problem was considered in a series of papers [2], [22], and [24] by Meir and Moon. In [2], the limiting distribution of W n is given for simply generated trees and it turns out that W n tends to the negative binomial distribution of order two, thus P{W n = m} mp 2 p m, for certain constants 0 < p < and n. For the special instance of Cayley trees they are simply generated trees with degree generating function ϕt = e t, this result was already obtained in [24]. In [22], W n was studied for recursive trees and for unordered unlabelled rooted trees, where asymptotic results for the first moments were given. The same problem has been studied recently also when analyzing the size of incomplete or one-sided trees for certain tree families. For binary search trees, the distribution of W n was given by Mahmoud [5] and limiting distribution results for W n are obtained by Itoh and Mahmoud in [2] for so called interval trees see definitions there. Quite recently obtained results for W n by the author are given below Limiting distribution for binary increasing trees and binary search trees. Theorem 7. [30] For binary increasing trees and also for ordinary binary search trees, the distribution of the random variable W n, which counts the number of steps that are used to climb a random tree of size n, is for n asymptotically Gaussian, where the rate of convergence is of order O log n : { } Wn log n P = Φx + O. log n log n

10 0 A. PANHOLZER Limiting distribution for recursive trees. Theorem 8. [30] For recursive trees, the distribution of the random variable W n, which counts the number of steps that are used to climb a random tree of size n, is for n asymptotically Gaussian, where the rate of convergence is of order O { } Wn log log n P = Φx + O. log log n log log n log log n : Limiting distribution for heap ordered trees. Theorem 9. [30] For heap ordered trees, the random variable W n, which counts the number of steps that are used to climb a random tree of size n, converges for n in distribution to a discrete random variable W, which has the distribution P{W = m} = m, for m 0. m +! Limiting distribution for polynomial increasing trees. Theorem 0. [30] Let ϕt = ϕ 0 + ϕ t + + ϕ d t d with ϕ 0 > 0, ϕ d > 0 and ϕ k 0 for 0 < k < d be a polynomial with degree d 2. Then for increasing trees with degree generating function ϕt, the distribution of the random variable W n, which counts the number of steps that are used to climb a random tree of size n, is for n asymptotically Gaussian, where the rate of convergence is of order O log n : W n log n d P log n = Φx + O. log n d Limiting distribution for unordered unlabelled rooted trees. Theorem. [30] For unordered unlabelled rooted trees, the random variable W n, which counts the number of steps that are used to climb a random tree of size n, converges for n in distribution to a discrete random variable W. Furthermore, the probabilities P{W = m} with m N 0 behave asymptotically like with certain constants K and α. P{W = m} = Km α m + O, m 2.3. Results concerning the cutting-down procedure. This problem was studied first by Meir and Moon in [9] for Cayley trees, where the first two moments of Z n are given exact and asymptotically. Chassaign and Marchand used in [3] an interpretation in terms of hashing tables to obtain the limit law of Z n for Cayley trees. For recursive trees, Meir and Moon gave in [20] asymptotic results for the first two moments of Z n. Quite recently obtained results for Z n by the author are given below.

11 ANALYSIS OF SOME TREE STATISTICS Limiting distribution for very simple trees. Our approach only works for certain simply generated tree families. They are defined by their degree generating functions ϕt in Lemma and are there called very simple trees. Theorem 2. [29] Let ϕt = k 0 ϕ kt k a function as defined in Lemma and τ denotes the unique solution of the equation ϕt = tϕ t of smallest modulus. Then for simply generated tree families with degree generating function ϕt and thus for very simple tree families, the distribution of the Z n, which counts the number of random cuts that will be done to destroy a randomly selected tree of size n, is for n asymptotically Rayleigh distributed: Z n n d Z, where Z is a random variable with density function fx = ϕ τx ϕ τx 2 τϕ τ e 2τϕ τ, for x 0, and fx = 0 otherwise r-th moments and r-th centered moments for recursive trees. Theorem 3. [28] For recursive trees, the r-th moments E Zn r resp. r-th centered moments E [Z n EZ n ] r of the random variable Z n, which counts the number of random cuts that will be done to destroy a randomly selected tree of size n are, for r an integer and n, asymptotically given by EZn r = nr log r n + r + H r γr n r log r+ n + O n r log r+2, for r, n E [Z n EZ n ] r = r r r n r log r+ n + O Limiting distribution for non-crossing trees. n r log r+2 n, for r 2. Theorem 4. [27] For non-crossing trees, the distribtion of the random variable Z n, which counts the number of random cuts that will be done to destroy a randomly selected tree of size n, is for n asymptotically Rayleigh distributed: Z n n d Z, where Z is a random variable with density function fx = 2 x2 xe 3, for x 0, and fx = 0 otherwise Proof sketches for the given results All given results are obtained by using a generating functions approach where the recursive description of the analyzed tree parameters was translated into equations mainly differential and functional equations for suitable generating functions. When using such a generating functions approach, the concept of singularity analysis see [0] is of great importance. These are transfer lemmata O-, o- and -transfers that allow to transfer the local behaviour of the generating functions around its dominant singularities to the asymptotic behavour of its coefficients.

12 2 A. PANHOLZER To obtain distribution results, a certain central limit theorem the so called quasi power theorem which is due to Hwang, see [] is very powerful, in particular when dealing with combinatorial structures. The quasi power theorem as proven in [] is given below, since it has been applied very frequently to obtain our stated results. Theorem 5. [H. K. Hwang] Let {Ω n } n be a sequence of integral random variables. Suppose that the moment generating function satisfies the asymptotic expression M n s := E e Ω ns = m 0 the O term being uniform for s σ, s C, σ > 0, where P{Ω n = m}e ms = e H ns + Oκ n, i H n s = Usφn + V s, with Us and V s analytic for s σ and independent of n; U 0 0, ii φn, iii κ n. Under these assumptions, the distribution of Ω n is asymptotically Gaussian: { Ωn U } 0φn P < x = Φx + O +, U 0φn κ n φn uniformly with respect to x, x R. Moreover, the mean and the variance of Ω n satisfy EΩ n = U 0φn + V 0 + Oκ n, VΩ n = U 0φn + V 0 + Oκ n. 3.. Proof sketches concerning the ancestor-tree size and the Steiner-distance see [25, 27, 3, 32, 34, 35]. We use a recursive appraoch, where the recursive description of the ancestor-tree size resp. the Steiner-distance is translated into functional and differential equations for generating functions Gz, u, v for the probabilities P{X n,p = m} resp. F z, u, v for the probabilities P{Y n,p = m} Simply generated trees: Introducing trivariate generating functions Gz, u, v = P{X n,p = m} n p Tn z n u p v m and F z, u, v = n 0 p n m 0 n m} n p Tn z n u p v m, we obtain the functional equations and 0 p n m 0 P{Y n,p = Gz, u, v = zv + uϕ Gz, u, v + vt z, 8 F z, u, v = Gz, u, v zvϕ T z Gz, u, v + zϕ T z F z, u, v z vϕ T z T z. 9 From the functional equation 8 one can show inductively for fixed p the representation [u p ]Gz, u, v = 2p k= α p,k z vz vzϕ T z k, 0 with functions α p,k z, that are analytic in the disk z ρ except at z = ρ if d = and thus ϕt is non-periodic, but the general case d > can be treated analogously. Here the expansion α p,k z = α p,k + O ρ z holds for an α p,k C in a neighborhood of z = ρ. From 0 one can extract the coefficients [z n v m ] asymptotically for n and m C n

13 ANALYSIS OF SOME TREE STATISTICS 3 for a fixed C > 0 by the Cauchy integration formula for a Hankel contour like integration path and obtains eventually the part of Theorem concerning X n,p. From equations 9 and 0, we obtain for fixed p 2 the representation [u p ]F z, u, v = 2p k= vzα p,k z zϕ T z vz k vzϕ T z, with the α p,k z as in 0 and one shows again by the Cauchy integration formula the remaining part of Theorem Recursive trees: Introducing trivariate generating functions Gz, u, v = P{X n,p = m} n z n p n up v m and F z, u, v = n 0 p n m 0 n m} n z n p n up v m, we obtain the differential equations 0 p n m 0 P{Y n,p = z Gz, u, v = v + uegz,u,v + v z, 2 and F z, u, v = z z F z, u, v + v v Gz, u, v Gz, u, v z z z log z. 3 Equation 2 can be solved exactly and we get Gz, u, v = log z v + u z v u, 4 which gives easily for fixed p the following uniform expansion around v = : [u p ]Gz, u, v = p z pv + O z p v. Singularity analysis and the quasi power theorem proves then the part of Theorem 2 concerning X n,p. Solving the differential equation 3 for F z, u, v gives F z, u, v = v log z + z + Gz, u, v + v z z t=0 Gt, u, v dt, and by using the exact solution 4 for Gz, u, v one shows again by singularity analysis and using the quasi power theorem the remaining part of Theorem Heap ordered trees: Introducing trivariate generating functions Gz, u, v = P{X n,p = m} n z n p Tn n! up v m and F z, u, v = n 0 p n m 0 n m} n z n p Tn n! up v m, we obtain the differential equations 0 p n m 0 P{Y n,p = v + u Gz, u, v = z Gz, u, v + v, 5 2z and z F z, u, v = z Gz, u, v+f z, u, v 2z Gz, u, v v 2z v 2z 2z. 6

14 4 A. PANHOLZER From 5 one can show by induction for fixed p the following expansion, which holds uniformly around v = : [u p g p v log 2z ]Gz, u, v = + O, 7 2z pv+ 2 2z p v+ 2 with functions g p v analytic around v =. The part of Theorem 3 concerning X n,p follows from this expansion after appying singularity analysis and the quasi power theorem. Studying 6 gives by using 7 for fixed p 2 the uniform expansion around v = [u p ]F z, u, v = g pvp v + pv + 2 2z pv+ 2 from which the remaining part of Theorem 3 follows. + O log 2z 2z p v Polynomial increasing trees: Introducing trivariate generating functions Gz, u, v = P{X n,p = m} n z n p Tn n! up v m and F z, u, v = n 0 p n m 0 n m} n z n p Tn n! up v m, we obtain the differential equations and F z, u, v = z 0 p n m 0, P{Y n,p = z Gz, u, v = v + u ϕ Gz, u, v + v ϕ T z, 8 ϕ T z F z, u, v + z Gz, u, v vϕ T z Gz, u, v vϕ T z T z. 9 From 5 one can show by induction for fixed p the following expansion, which holds uniformly for v σ, σ > 0: [u p ]Gz, u, v = g p v z pdv d + p d + O z pd d + p d 3p 2dσ d, 20 ρ ρ with functions g p v analytic around v =. The part of Theorem 4 concerning X n,p follows from this expansion after appying singularity analysis and the quasi power theorem. From 9 and 20 we obtain for fixed p 2 the uniform expansion around v = [u p ]F z, u, v = g pvp dv z pdv d + p d + O pdv p d + ρ and the remaining part of Theorem 4 follows. z ρ pd d + p d 3p 2dσ d Generalized M-ary search trees: With trivariate generating functions Gz, u, v = n n 0,p 0,m 0 p P{XM,t;n,p = m}z n u p v m and F z, u, v = n n 0,p 0,m 0 p P{YM,t;n,p = m}z n u p v m one obtains the nonlinear differential equation Mt +! Mt+ v + um Gz, u, v = zmt+ t! M t Gz, u, v zt M + v z Mt+, 2

15 and the Euler differential equation ANALYSIS OF SOME TREE STATISTICS 5 Mt+ M t F z, u, v = Mt +! zmt+ t! z M t+ F z, u, v zt 22 Mt+ M v t M v + Gz, u, v Mt +! zmt+ t! z M t+ Gz, u, v zt z Mt+. From 2 one can show by induction for fixed p the following expansion, which holds uniformly around v = : [u p ] t g Gz, u, v = zt + O p v z pλ v p t+ z p λ v p t++λ 2 v + O log z z p λ, v p 2t+ with functions g p v analytic around v =. The appearing λ i v are the roots of the indicial polynomial p M,t λv = λv τ t + 2 τ v, 24 arranged by descending order of real parts, where we used the notation x k := x x+ x+ k for the rising factorials and the abbreviation τ := M t +. Singularity analysis and the quasi power theorem show then the part of Theorem 5 concerning X M,t;n,p From equations 22 and 23 one obtains for fixed p 2 the following uniform expansion around v = : [u p ] t f F z, u, v = zt + O p v z pλ v p t+ z p λ v p t++λ 2 v + O log z z p λ, v p 2t+ with functions f p v analytic around v =. Again, via singularity analysis and applying the quasi power theorem, one shows the remaining part of Theorem Non-crossing trees: With generating functions Gz, u, v = n 0 p n m 0 T n n p P{Xn,p = m}z n u p v m, F z, u, v = n 0 p n m 0 T n n p P{Yn,p = m}z n u p v m and auxiliary functions Gz, u, v resp. F z, u, v, we obtain the following systems of functional equations: Gz, u, v = and F z, u, v = Gz, u, v F z, u, v = Gz, u, v 23 zv + u Gz, u, v + vt z, zv + u Gz, u, v = Gz, + vbz 25 u, v 2 zv Bz Gz, z 2 u, v + Bz F z vbz 2 z, u, v Bz 2, 2zv Bz 3 Gz, u, v + 2z Bz 3 F z, u, v 2z vbz Bz Studying these functional equations in a way similar as done for simply generated trees leads to the given Theorem 6.

16 6 A. PANHOLZER 3.2. Proof sketches concerning the climbing procedure see [30]. We use a recursive approach, where the recursive nature of the climbing procedure is translated into recurrences for the random variables W n. These can be treated via generating functions: in the instance of increasing tree families, one uses bivariate generating functions W z, v = n m 0 T np{w n = m} zn n! vm and obtains the linear first order differential equation ϕ T z ϕ0 W z, v = v W z, v + ϕ 0, W 0, v = 0. z T z This equation has the solution W z, v = ϕ 0 e v z ϕt t ϕ 0 u dt z T t t=α e v t=α u=0 ϕt t ϕ 0 T t dt du, 27 where α can be chosen arbitrary, such that the integral is defined. In the instance of unordered unlabelled rooted trees, one uses bivariate generating functions W z, v = n m 0 T np{w n = m}z n v m and obtains the functional equation W z, v = z + zv j W z j, v k 0 Z k {T z}. 28 k + j Introducing for 0 t the functions fz, t := z Z k {T z}t k = z exp k 0 k T z k t k k and gz, v, t := j 2 W z j, vt j, we can write the solution of the functional equation 28 as W z, v = z + v t=0 fz, tgz, v, tdt v t=0 fz, tdt Binary increasing trees and binary search trees: For binary increasing trees, we obtain by specializing ϕt = + t 2 in 27 the generating function W z, v = e zv z v t v e tv dt. t=0 Studying the asymptotic behaviour of W z, v around the dominant singularity z = uniformly in a neighbourhood of v =, we obtain the uniform expansion W z, v = Cv z v + O z v, with Cv := t=0 tv e tv dt. Singularity analysis and applying the quasi power theorem immediately gives Theorem 7.

17 ANALYSIS OF SOME TREE STATISTICS Recursive trees: For recursive trees, we obtain by specializing ϕt = e t in 27 the generating function W z, v = e z t v t=α t log t dt z u=0 e u t v t=α t log dt t du, where it is here advantageous to choose α := e. Here we obtain for W z, v when expanding around the dominant singularity z = uniformly in a neighbourhood of v = the expansion W z, v = log v v e z t=α log t dt u=0 e u t v t=α t log dt t du + O z. This is sufficient to show via singularity analysis and the quasi power theorem the given Theorem Heap ordered trees: For heap ordered trees, we obtain by specializing ϕt = t in 27 the generating function W z, v = v 2z v 2 + v v 2 e v 2z. Expanding around the dominant singularity z = 2 uniformly in a neighbourhood of v = gives W z, v = ev v + + v e v v 2z v 2 2z + + O 2z 3 2. v 2 Singularity analysis shows then that the moment generating function Ee Wns = m 0 P{W n = m}e ms of W n converges uniformly in a neighbourhood of s = 0 to the moment generating function Ee W s = +es e es e of a discrete random variable W, with s distribution P{W = m} =, for m 0, and thus Theorem 9 holds. m m+! Polynomial increasing trees: Here one can show by considering 27 in more detail the following asymptotic expansion of W z, v around the dominant singularity z = ρ, which holds uniformly around v = : W z, v = Cv z ρ v d + O z ρ d, with a function Cv that is analytic around v =. From this expansion one shows via singularity analysis and applying the quasi power theorem the stated Theorem Unordered unlabelled rooted trees: Here one can show by studying 29 the following expansion of W z, v which holds uniformly around v = in a neighbourhood of the dominant singularity z = ρ: W z, v = av bv ρ z + Oρ z, with certain functions av and bv, analytic around v =. Singularity analysis gives that the moment generating function Ee W ns of W n converge uniformly in a neighbourhood of s = 0 to the moment generating function Ee W s of a discrete random variable W, with distribution P{W = m} = [v m ]bv, for m 0.

18 8 A. PANHOLZER Moreover by examining bv in detail, one can show the expansion [v m ]bv = Km α m + O, m with certain constants K and α, and thus Theorem holds Proof sketches concerning the cutting down procedure see [27, 28, 29]. In principle one uses a recursive approach, where one studies the random variable Z n by treating the recurrence n P{Z n = m} = p n,k P{Z k = m }, for n 2, m, 30 k= with initial values P{Z = 0} = and P{Z n = 0} = 0 for n 2. Here p n,k denotes the probability, that by choosing a random tree of size n from the given tree family and removing a random edge, the remaining subtree containing the root, is of size k. To reduce for a certain tree family the original problem to the study of above recurrence 30, it is necessary, that randomness is preserved by cutting off a random edge, what means, that starting with a random tree of size n and removing a random edge, the remaining subtree of size k containing the root is actually a random tree of size k in this tree familiy. Non-crossing trees resp. recursive trees preserve randomness by cutting off a random edge. This does not hold for general simply generated tree families; only a sublass of them have this property and this is here called the class of very simple tree families. These trees can be characterized by the degree generating functions ϕt: Lemma. A simply generated tree family preserves randomness when cutting off a random edge and is thus a very simple tree family iff the degree generating function ϕt = k 0 ϕ kt k is given by one of the following three formulæ. with ϕ 0 > 0, α 0 > 0, 2α α 0 > 0. Case A : ϕt = ϕ 0 e α0t, Case B : ϕt = ϕ 0 + α d 0t, d 2, d ϕ 0 Case C : ϕt = 2α α 0 t α 0, 2α α Very simple trees: Here, the probabilities p n,k that occur in 30 are given as: Case A : Case B : Case C : p n,k = α 0k T k T n k n T n, p n,k = α 0kd + T k T n k dn T n, p n,k = 2α k 2α + α 0 T k T n k n T n. For all three cases it holds that by introducing the bivariate generating function Mz, v = 0 m n T n P{Z n = m}z n v m, the recurrence 30 can be translated into linear first n

19 ANALYSIS OF SOME TREE STATISTICS 9 order differential equations which can be solved similarly. We give only the equation for Case C: z Mz, v = 2α α 0 vt z, Mz, v z 2α vt z which gives after adapting to the initial values the solution: 2α α 0 T z α 0 v 2α v 2α α 0 Mz, v = T z. 2α vt z Using singularity analysis, one can obtain from this explicit solution the asymptotic behaviour of the r-th factorial and ordinary moments, which gives, together with the Theorem of Fréchet and Shohat see e. g. [7, p. 242], eventually Theorem Recursive trees: Here, the probabilities p n,k that occur in 30 are given as: n p n,k = n n kn k +. The distribution recurrence 30 leads to recurrences for the r-th moments E Zn r, which can be treated via generating functions M [r] z = n E Zn r z n n. One obtains the differential equations d z log z dz M [r] z M [r] z = S [r] z, 3 where S [r] z is given by r S [r] z = l=0 r d l dz M [l] z z z log Solving and adapting to the initial condition leads to the solution M [r] z = log z z t=0. z S [r] t t log 2 32 t tdt. From this explicit solution one can show inductively the asymptotic growth of the ordinary moments E Z r n as given in Theorem 3. The given result for the centered moments E [Zn EZ n ] r follows by applying summation formulæ Non-crossing trees: Here, the probabilities p n,k that occur in 30 are given as: p n,k = 3k 22n 3k 3 3n k 2 k n k n 2k n k 3n 3 n Introducing the bivariate generating function Mz, v = n the recurrence 30 can be translated into the linear first order differential equation z Mz, v = 2vBz Mz, v z 3vBz.. Adapting to the initial values leads to the following solution for Mz, v: v Bz 3v Mz, v = Bz Bz. 3vBz m 0 T np{z n = m}z n v m,

20 20 A. PANHOLZER Again by using singularity analysis, one can obtain from this explicit solution the asymptotic behaviour of the r-th moments, which gives, together with the Theorem of Fréchet and Shohat, eventually Theorem 4. References [] D. Aldous, The continuum random tree II: an overview. Stochastic analysis Proc., Durham, 990, 23 70, London Math. Soc. Lecture Note Ser. 67, Cambridge Univ. Press, Cambridge, 99. [2] F. Bergeron, F. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science 58, 24 48, 992. [3] P. Chassaing and R. Marchand, Cutting a Random Tree and UNION-FIND Algorithms, Talk at the 8th Seminar on the Analysis of Algorithms, Strobl, [4] H.-H. Chern and H.-K. Hwang, Phase changes in random m-ary search trees and generalized quicksort, Random Structures and Algorithms 9, , 200. [5] L. Devroye, Applications of the theory of records in the study of random trees, Acta Informatica 26, 23 30, 988. [6] R. Dobrow, On the distribution of distances in recursive trees, Journal of Applied Probability 33, , 996. [7] M. Fisz, Probability Theory and Mathematical Statistics, John Wiley, New York, 963. [8] P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Mathematics 204, , 999. [9] Ph. Flajolet and A. Odlyzko, The Average Height of Binary Trees and Other Simple Trees, Journal of Computer and System Sciences 25, 7 23, 982. [0] P. Flajolet and A. Odlyzko, Singularity Analysis of Generating Functions, SIAM Journal of Discrete Mathematics 3, , 990. [] H.-K. Hwang, On Convergence Rates in the Central Limit Theorems for Combinatorial Structures, European Journal of Combinatorics 9, , 998. [2] Y. Itoh and H. Mahmoud, One-sided variations on interval trees, Journal of Applied Probability 40, , [3] H. Mahmoud, Evolution of Random Search Trees, Wiley, New York, 992. [4] J. Mahmoud, Distances in random plane-oriented recursive trees, Journal of Computational and Applied Mathematics 4, , 992. [5] H. Mahmoud, One-sided variations on binary search trees, Annals of the Institute of Statistical Mathematics 55, , [6] H. Mahmoud and R. Neininger, Distribution of Distances in Random Binary Search Trees, The Annals of Applied Probability 3, , [7] H. Mahmoud and B. Pittel, On the joint distribution of the insertion path length and the number of comparisons in search trees, Discrete Applied Mathematics 20, , 988. [8] H. Mahmoud and R. Smythe, A Survey of Recursive Trees, Theoretical Probability and Mathematical Statistics 5, 37, 995. [9] A. Meir and J. W. Moon, Cutting down random trees, Journal of the Australian Mathematical Society, , 970. [20] A. Meir and J. W. Moon, Cutting down recursive trees, Mathematical Biosciences 2, 73 8, 974. [2] A. Meir and J. W. Moon, Climbing certain types of rooted trees I, Proceedings of the Fifth British Combinatorial Conference, , 975. [22] A. Meir and J. W. Moon, Climbing certain types of rooted trees II, Acta Mathematica Academia Scientiarum Hungaricae 3, 43 54, 978. [23] A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian Journal of Mathematics 30, , 978. [24] J. W. Moon, Climbing random trees, Aequationes Mathematicae 5, 68 74, 970. [25] K. Morris, A. Panholzer and H. Prodinger, On some parameters in Heap ordered trees, accepted for publication in Combinatorics, Probability and Computing. [26] R. Otter, The number of trees, Annals of Mathematics 49, , 948.

21 ANALYSIS OF SOME TREE STATISTICS 2 [27] A. Panholzer, Non-crossing trees revisited: cutting down and spanning subtrees, Discrete Mathematics and Theoretical Computer Science, [28] A. Panholzer, Destruction of recursive trees, accepted for publication in Proceedings of the Third Colloquium on Mathematics and Computer Science, Birkhäuser, [29] A. Panholzer, Cutting down very simple trees, submitted. [30] A. Panholzer, Climbing rooted trees again, submitted. [3] A. Panholzer, The distribution of the size of the ancestor tree and of the induced spanning subtree for random trees, accepted for publication in Random Structures and Algorithms. [32] A. Panholzer, Distribution of the Steiner Distance in Generalized M-ary Search Trees, accepted for publication in Combinatorics, Probability and Computing. [33] A. Panholzer and H. Prodinger, A generating functions approach for the analysis of grand averages for multiple QUICKSELECT, Random Structures Algorithms 3, , 998. [34] A. Panholzer and H. Prodinger, Spanning tree size in random binary search trees, The Annals of Applied Probability 4, , [35] A. Panholzer and H. Prodinger, Analysis of some statistics for increasing tree families, submitted. [36] G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica 68, , 937. Alois Panholzer, Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8-0/04, 040 Wien, Austria address: Alois.Panholzer@tuwien.ac.at