Varieties of Increasing Trees

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a uniied generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary diierential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymp-totically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n logn), the distributions of node levels and number of leaves are asymptotically normal, etc. Vari et es d'arbres croissants R esum e. Un arbre croissant est un arbre enracin e et etiquet e dans lequel les etiquettes le long de toute branche issue de la racine vont en ordre croissant. Sous des d eguisements divers, de tels arbres sont apparus comme repr esentations arborescentes des permutations, comme structures de donn ees en informatique et comme mod eles probabilistes dans diverses applications. L'on pr esente ici une approche unii ee fond ee sur les s eries g en eratrices a l'analyse de param etres sur de tels arbres. Les s eries enum eratives sont li ees etroitement a une equation dii erentielle autonome et non lin eaire. L'analyse de singularit es s'applique naturellement a de telles s eries, ce qui permet l'analyse asympto-Abstract An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a uniied generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary diierential equation, d dz Y (z) = (Y (z)); which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of …