For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let |n | denote the binary length of a natural number n, let |n|h denote the h-times iterated binary length of n and let inv(n) be the least h such that |n|h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0,..., αn 〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + |i | · |i|h. As a supplement to this (refined Friedman style) independence result we further show that e.g. primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0,..., αn 〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K +|i|· inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations. Using results from Otter and from Matouˇsek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter’s tree constant 2.9557652856.... ∗ Research supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft. † The main results of this paper were obtained during the authors visit of T. Arai in

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k; 4 k #n - 2). In this paper, we investigate the behavior of the diameter of MST in randomly-weighted complete graphs (in Erds-Rnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is small---independent of n, we present a one-time-tree-construction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...

Erdos, P. L., A new bijection on rooted forests, Discrete Mathematics 111 (1993) 179-188. This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees. A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). This method yields new easy proofs for some well-known theorems which use only elementary calculations with the sum of stirling numbers.

ABSTRACT. 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.

A queue-based Prufer-like code is used to determine the expected number of level-i nodes in a random labeled tree on n nodes. Level-1 nodes are the leaves of a given tree and level-i nodes are leaves after all nodes in levels 1 through (i-1) have been deleted. More precisely, we study the expected fraction f(i) of n nodes that are in levels 1 through i. Tight bounds on f(i) are obtained and used to estimate the expected radius of a random labeled tree.