Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Singularity analysis" reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complex-analytic Tauberian procedure by which combinatorial constructions and asymptotic-probabilistic laws can be systematically related.

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Introduction The rapid growth of programs produced by genetic programming (GP) is a well documented phenomenon [Koza, 1992; Blickle and Thiele, 1994; Nordin and Banzhaf, 1995; McPhee and Miller, 1995; Soule et al., 1996; Greeff and Aldrich, 1997; Soule, 1998] . This growth, often referred to as "code bloat", need not be correlated with increases in the fitness of the evolving programs and consists primarily of code which does not change the semantics of the evolving program. The rate of growth appears to vary depending upon the particular genetic programming paradigm being used, but exponential rates of growth have been documented [Nordin and Banzhaf, 1995] . Code bloat occurs in both tree based and linear genomes [Nordin, 1997; Nordin and Banzhaf, 1995; Nordin et al., 1997] and with automatically defined functions [Langdon, 1995] . Recent research suggests that code bloat will occur in most fitness based search techniques which allow variable length solutions [Langdon, 1998b; Langdo

Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I,..., n. It is shown that HJog n + c = 4.3 I 107... in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p> 0, lim,,E(H$)/ log % = cp. Finally, it is proved that &/log n--, c * = 0.3733..., in probability, where c * is defined by c log((2e)lc) = 1, c 5 1, and.S, is the saturation level of the same tree, that is, the number of full levels in the tree.

Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.

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Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random k-d trees, quadtrees and union-end trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that H„/(c log (n)) tends to 1 in probability and in the mean as n- oo, where H „ is the height of the tree, and c =4.31107... is a solution of the equation c log (2e / = 1. In addition, we ~c ~ show that H „-clog (n) = O (/log (n) loglog (n)) in probability.

In earlier work we predicted program size would grow in the limit at a quadratic rate and up to fifty generations we measured bloat O(generations 1:2\Gamma1:5 ). On two simple benchmarks we test the prediction of bloat O(generations 2:0 ) up to generation 600. In continuous problems the limit of quadratic growth is reached but convergence in the discrete case limits growth in size. Measurements indicate subtree crossover ceases to be disruptive with large programs (1,000,000) and the population effectively converges (even though variety is near unity). Depending upon implementation, we predict run time O(no. generations 2:0\Gamma3:0 ) and memory O(no. generations 1:0\Gamma2:0 ). 1 INTRODUCTION It has been known for some time that programs within GP populations tend to rapidly increase in size as the population evolves [ Koza, 1992, Altenberg, 1994, Tackett, 1994, Blickle and Thiele, 1994, Nordin and Banzhaf, 1995, Nordin, 1997, McPhee and Miller, 1995, Langdon,...

Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton-- Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.

Abstract. We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random, and the case when we condition on the tree. The proofs are based on Aldous ’ theory of the continuum random tree. 1.