Planted Plane Trees Research Articles

The average height of planted plane trees with M leaves

The number of planted plane trees with n nodes, m leaves, and height h is computed. Assuming that all n-node planted plane trees with m leaves are equally likely, it is shown that the average height h(n, m) is asymptotically given for all ε > 0 and fixed ϱ = m n , 0 < ϱ < 1, by h(n,m)=(φn(ρ -1 − 1)) 1 2 + 1.5 − ρ -1 + O( ln(n) n 1 2-ε )

A note on a result of R. Kemp on R-typly rooted planted plane trees

R. Kemp has shown that the average height of r-tuply rooted planted plane trees is $$\sqrt {\pi n} - \frac{1}{2}(r - 2) + O(\log (n)n^{1/2 - \varepsilon } ), \varepsilon > 0, n \to \infty ,$$ assuming that all such trees withn nodes are equally likely. We give a quite short proof of this result (with an error term ofO (1)).

On the average oscillation of a stack

Evaluating a binary tree in postorder we assume that in one unit of time zero or two nodes are removed from the top of the stack and one node is stored in the stack. The oscillation of the stack can be described by a functionf wheref(t) is the number of nodes in the stack aftert units of time. In this paper we shall first derive several new enumeration results concerning planted plane trees. In the second part we shall prove, that the average number of maxima (MAX-turns) and minima (MIN-turns) of the functionf isn/2 andn/2—1, respectively, provided that all binary trees withn leaves are equally likely. Finally, we shall compute for largen and fixedj the average increase (decrease) of the length of the stack between thej-th MIN-turn and (j+1)-th MAX-turn (between thej-th MAX-turn and thej-th MIN-turn). This result implies that the average oscillation of the stack can be described by the functionf(j)=4√j/π−(−1) j +O(1/√j) for largen and fixed turn-numberj.

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα s (j), which is of orderj 1/2. Thej-th ”s-hyperoscillation”α 1(j)−α s+1(j) is given by 1/2α 1(s)+O(j −1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s 1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Polyamethod.

The average height of r-typly rooted planted plane trees

In this paper we generalize a result of de Bruijn, Knuth und Rice concerning the average height of planted plane trees withn nodes. First we compute the number of allr-typly rooted planted plane trees (r-trees) withn nodes and height less than or equal tok. Assuming that all planted plane trees withn nodes are equally likely, we show, that in the average a planted plane tree is a 3-tree for largen; for this distribution we compute also the cumulative distribution function and the variance. Finally, we shall derive an exact expression and its asymptotic equivalent for the average height\(\bar h_r \) (n) of anr-tree withn nodes. We obtain for all e>0 $$\bar h_r (n) = \sqrt {\pi n} - \frac{1}{2}(r - 2) + O(1n(n)/n^{1/2 - \varepsilon } ).$$

Enumeration of coloured plane trees with a given type partition

Tutte's result for the number of planted plane trees with a given degree partition is rederived by a variety of methods and in particular by a simple piecewise construction technique. A theorem of Gordon and Temple is applied in order to give a general relationship between the number of planted plane trees and the number of rooted plane trees and the degree partition restriction is generalised to type partition. The piecewise construction method is successfully used to derive the number of planted plane trees with a given 2-colour degree partition, also derived by Tutte, and an algorithm for the k-coloured case is developed. This algorithm may be used to obtain more specific results. These models are relevant to the statistical mechanics of polymers and this is discussed briefly.

A combinatorial interpretation for two transformations of series that commute with compositional inversion

For a formal power series g(t) = 1 [1 + ∑ n=1 ∞h nt n] with nonnegative integer coefficients, the compositional inverse f(t) = t · f(t) of g(t) = t · g(t) is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of h i colors. Since the compositional inverse of the Euler transformation of f( t) is the star transformation [[ g( t)] −1 − 1] −1 of g( t), [2], it follows that the Euler transformation of f( t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of h i colors for i > 1, and h 1 − 1 colors for i = 1.

Achiral plane trees

AbstractHarary and Robinson showed that the number an of achiral planted plane trees with n points coincides with the number pn of achiral plane trees with n points, for n ⩾ 2. They posed the problem of finding a natural structural correspondence which explains this coincidence. In the present paper this problem is answered by constructing two‐to‐one correspondences from certain sets of binary sequences to each of the sets of trees concerned, giving a structural basis for the equation 2an = 2pn. Answers are also supplied to similar correspondence‐type problems of Harary and Robinson, concerning planted plane trees, and achiral rooted plane trees. In addition, each of these four types of plane trees are counted with numbers of points and endpoints as the enumeration parameters. The results all show a symmetry with respect to the number of endpoints which is not shared by the set of all plane trees.

On the enumeration of certain sets of planted plane trees

Using the definition of planted plane trees given by D. A. Klarner (“A correspondence between sets of trees,” Indag. Math. 31 (1969), 292–296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of ( r + 1)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.

Valency enumeration of rooted plane trees

AbstractWe obtain the enumerator by node-valencies of planted plane trees, whose square gives the enumerator of rooted plane trees. We also study the enumeration by number of nodes and black-node-valencies of bichromatic rooted plane trees, encountering a remarkably simple inversion formula. Finally, we remark that these bichromatic trees are in 1–1 correspondence with solutions to the weak lead ballot problem.

Ballots and plane trees

The correspondence of certain plane trees and binary sequences reported by D. A. Klarner in [1], and a ballot interpretation of the latter, are used to make an independent evaluation of the number of classes of isomorphic, ( k + 1)-valent, planted plane trees with kn + 2 points. This provides an interesting multivariable identity for binomial coefficients.

Correspondences between plane trees and binary sequences

The subject of each of the five sections of this paper is the planted plane trees discussed by Harary, Prins, and Tutte [7]. A description of the content of the present work is given in Section 1. Section 2 is devoted to a definition of plane trees in terms of finite sets and relations defined on them—we hope this definition will replace the topological concepts introduced in [7]. A one-to-one correspondence between the classes of isomorphic planted plane trees with n+2 vertices and the classes of isomorphic 3-valent planted plane trees with 2 n+2 vertices is given in Section 3. Sections 4 and 5 deal with enumeration problems.

Languages and the enumeration of planted plane trees

This paper sets into relation three different classes of trees: (i) mod r-valent planted plane trees; (ii) r-valent planted plane trees; and (iii) subtrees of a certain infinite planted tree. Practically, all three classes of trees are counted by the same generating function, which is calculated by mapping trees on the words of a language. This leads to producing a one-to-one correspondence between the three classes by means of codes.

A note on plane trees

The paper gives three different ways of producing the one-to-one correspondence between planted plane trees and trivalent planted plane trees discovered by Harary, Prins, and Tutte.

The number of plane trees with a given partition

This note is a continuation of the articles [6] and [2]. In [1], trees with a given partition α = ( a 1 ; a 2 , …), where a i is the number of vertices (points) of valency (degree) i were enumerated. After the determination of the number of plane trees in [2], the number of planted plane trees with a given partition α was found explicitly in [6]. In the present note, the number of plane trees with a given partition is expressed as a function of the number of planted trees with a given partition. The method, which is not new, consists of an application of the enumeration techniques of Otter [3] and Polya [4]; it was used in [1] and also by Riordan [5].

The Number of Planted Plane Trees with a Given Partition

The Number of Planted Plane Trees with a Given Partition

The Number of Planted Plane Trees with a Given Partition

The Number of Planted Plane Trees with a Given Partition