Varieties of Increasing

---~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 nodes. In this way it is found that various models share common features: path length is O(n log n), the distribution of node levels and number of leaves are asymptotically normal, etc. I n t r o d u c t i o n A labelled tree of size n is a rooted tree comprising n nodes tha t are labelled by distinct integers of the set { 1 , . . . , n}. An increasing tree is a labelled tree such that the sequence of labels along any branch s tar t ing at the root is increasing. The enumeration of trees is a major branch of combinatorial analysis. A classical result due to Arthur Cayley in 1889 states tha t the number of labelled non-plane trees with n nodes is n n-1. For plane trees, the corresponding count is (n 1)! (~n..~2) since the number of unlabeUed plane 1 (~,~-2~ n! possible labellings given a fixed traversal trees is the Catalan number ~ ~ n-1 J and there axe order of trees, e.g., preorder. On this and other s tandard combinatorial analysis results, we refer the reader to the treatises of Comtet [6], Goulden and Jackson [19], or Bergeron et al. [3]. This paper concerns the enumeration of parameters on various families of increasing trees. The families to be considered axe of two types: (i) non-plane trees, which are taken in the graph theoretic sense so tha t subtrees stemming from a node are not ordered between themselves; (ii) plane trees, where a plane embedding is specified so that subtrees stemming from a node are ordered between themselves. Rather arbi t rary conditions can be imposed on the node degrees tha t axe allowed. D e f i n i t i o n 1 Let ~sr)~=0 be a sequence of non negative integers, such tha t so ~ 0 and Sr ~ 0 for some r ~_ 2. The variety of trees associated to ( s t} and the specification of v~n element of