Some results for monotonically labelled simply generated trees
نویسندگان
چکیده
We consider simply generated trees, where the nodes are equipped with weakly monotone labellings with elements of {1,2, . . . ,r}, for r fixed. These tree families were introduced in Prodinger and Urbanek (1983) and studied further in Kirschenhofer (1984), Blieberger (1987), and Morris and Prodinger (2005). Here we give distributional results for several tree statistics (the depth of a random node, the ancestor-tree size and the Steiner-distance of p randomly chosen nodes, the height of the j-st leaf, and the number of nodes with label l), which extend the existing results and also contain the corresponding results for unlabelled simply generated trees as the special case r = 1.
منابع مشابه
On Heights of Monotonically Labelled Binary Trees
A monotonically labelled binary tree is a binary tree with integer labels attached to the internal nodes in such a way that the labels along any path from the root are nondecreasing. If there is only one label, we are dealing with ordinary binary trees. Prodinger and Urbanek [6] introduced a variety of families of monotonically labelled trees and obtained enumerative results about many of them....
متن کاملMonotonically labelled Motzkin trees
Consider a rooted tree structure the nodes of which have been labelled monotonically by elements of { 1, 2, . . .,k}, which means that any sequence connecting the root of the tree with a leaf is weakly monotone . For fixed k asymptotic equivalents of the form CA gA °n; 2 (n --oo) to the numbers of such tree structures with n nodes are obtained for the family of extended unary-binary trees (i .e...
متن کاملAnalysis of three graph parameters for random trees
We consider three basic graph parameters, the node-independence number, the path node-covering number, and the size of the kernel, and study their distributional behaviour for an important class of random tree models, namely the class of simply generated trees, which contains, e.g., binary trees, rooted labelled trees and planted plane trees, as special instances. We can show for simply generat...
متن کاملThe Number of Descendants in Simply Generated Random Trees
The aim of this note is to generalize some recent results for binary trees by Panholzer and Prodinger [15] to a larger class of rooted trees. The number of descendants of a node j is the number of nodes in the subtree rooted at j, and the number of ascendants is the number of nodes between j and the root. Recently, Panholzer and Prodinger [15] studied the behavior of these parameters in binary ...
متن کاملSimply Generated Trees, Conditioned Galton–watson Trees, Random Allocations and Condensation: Extended Abstract
taking the product over all nodes v in T , where d+(v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let Tn be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w0 > 0 and that there exists some k > ...
متن کامل