The distribution of height and diameter in random non-plane binary trees

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height

The height of random binary unlabelled trees

This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.Comment: 14 page

The height of random binary unlabelled trees

This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height

The Shape of Unlabeled Rooted Random Trees

We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.Comment: 34 pages, 1 figur

Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability

XML Compression via DAGs

Unranked trees can be represented using their minimal dag (directed acyclic graph). For XML this achieves high compression ratios due to their repetitive mark up. Unranked trees are often represented through first child/next sibling (fcns) encoded binary trees. We study the difference in size (= number of edges) of minimal dag versus minimal dag of the fcns encoded binary tree. One main finding is that the size of the dag of the binary tree can never be smaller than the square root of the size of the minimal dag, and that there are examples that match this bound. We introduce a new combined structure, the hybrid dag, which is guaranteed to be smaller than (or equal in size to) both dags. Interestingly, we find through experiments that last child/previous sibling encodings are much better for XML compression via dags, than fcns encodings. We determine the average sizes of unranked and binary dags over a given set of labels (under uniform distribution) in terms of their exact generating functions, and in terms of their asymptotical behavior.Comment: A short version of this paper appeared in the Proceedings of ICDT 201

The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically