The cut‐tree of large trees with small heights

We describe a very simple selective search algorithm for two-player games, called best-first minimax. It always expands next the node at the end of the expected line of play, which determines the minimax value of the root. It uses the same information as alpha-beta minimax, and takes roughly the same time per node generation. We present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, but at significant time cost. Our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move, pruning subtrees below moves not chosen by either player. We also show how to efficiently generate a class of incremental random game trees. On uniform random game trees, best-first minimax outperforms alpha-beta, when both algorithms are given the same amount of computation. On random trees with random branching factors, best-first outperforms alpha-beta for shallow depths, but eventually loses at greater depths. We obtain similar results in the game of Othello. Finally, we present a hybrid best-first extension algorithm that combines alpha-beta with best-first minimax, and performs significantly better than alpha-beta in both domains, even at greater depths. In Othello, it beats alpha-beta in two out of three games.

We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees that we call probability fringe convergence, we show that the empirical spectral distributions for many random tree models converge to a deterministic (model-dependent) limit as the number of vertices goes to infinity.Moreover, the masses assigned by the empirical spectral distributions to individual points also converge in distribution to constants. We conclude for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain lower bounds on the mass assigned to zero by the empirical spectral measures via the connection between the number of zero eigenvalues of the adjacency matrix of a tree and the cardinality of a maximal matching on the tree. In particular, we employ a simplified version of an algorithm due to Karp and Sipser to construct maximal matchings and understand their properties. Moreover, we show that the total weight of a weighted matching is asymptotically equivalent to a constant multiple of the number of vertices when the edge weights are independent, identically distributed, nonnegative random variables with finite expected value, thereby significantly extending a result obtained by Aldous and Steele in the special case of uniform random trees.We greatly generalize a celebrated result obtained by Schwenk for the uniform random trees by showing that if any ensemble converges in the probability fringe sense and a very mild further condition holds, then, with probability converging to one, the spectrum of a realization is shared by at least one other (nonisomorphic) tree.For the linear preferential attachment model with parameter a>−1, we show that for any fixed k, the k largest eigenvalues jointly converge in distribution to a nontrivial limit when rescaled by \(n^{1/2\gamma_{a}}\), where γ a =a+2 is the Malthusian rate of growth parameter for an associated continuous-time branching process.

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.

We use large deviations to prove a general theorem on the asymptotic edge-weighted height Hn* of a large class of random trees for which Hn* ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees, random recursive trees and plane oriented trees for instance. New applications include the heights of some random lopsided trees and of the intersection of random trees.

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Cartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. If the search key and the priority key are independent, and the trees is built based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees. In this article, we analyze the expected behavior when the keys are dependent: in most cases, the expected search, insertion, and deletion times are Φ(√n). We indicate how these results can be used in the analysis of divide‐and‐conguer algorithms for maximal vectors and convex hulls. Finally, we look at distributions for which the expected time per operation grows like na for a ϵ[1/2, 1]. © 1994 John Wiley & Sons, Inc.

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. https://doi.org/10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

We find the asymptotic average composition of a class of nonclassic Polya urn models (not necessarily of fixed row sum) by embedding the discrete urn process into a renewal process with rewards. A subclass of the models considered has banded matrix urn schemes and serves as a natural modeling tool for the size of a class of random bucket trees. The class of urns considered extends known results for multicolor urns with constant row sums. The same asymptotic average results are shown to hold in the larger class. This provides an average-case analysis for the size of certain random bucketed multidimensional quad trees and k-d trees, which are all new results. Some bucket trees have urn schemes with constant row sum, a special case that helps detect phase changes in the limiting distribution of the (normed) size of the tree. For these special cases one can appeal to a more developed urn theory to find a limiting distribution of the normed size up to a threshold value of the capacity of a bucket. Once that cut-off point is surpassed, normality ceases to hold. This case appears in paged binary trees (threshold 116), m-ary search trees (threshold 26), and bucket recursive trees (threshold 26). The asymptotic normality results and the phase change after the threshold in these trees are already known and we only provide alternative proofs via a unified urn models approach.

We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.

We show that the variances of the profile (number of nodes at each level) of random binary search trees undergoes asymptotically four phase transitions and exhibits a bimodal or two-humped behavior, in contrast to the unimodality of the expected value of the profiles. Precise asymptotic approximations are derived. The same types of phenomena also hold for the profile of random recursive trees.

The theory of records in sequences of independent identically distributed random variables leads to simple proofs of various properties of random trees, including among other things, the limit law for the depth of the last node of random ordered trees, random union-find trees, and random binary search trees.

We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

On Robson's convergence and boundedness conjectures concerning the height of binary search trees

k-Protected vertices in binary search trees