Level number sequences for trees

This paper concerns some statistical properties of a parameter related to the profiles of binary trees. Define the level of a node v in a rooted tree t as the number of nodes on the branch connecting v to the root of t (counting both end nodes). The level number sequence of a tree t is the infinite sequence of integers (nl, n2, • • .) such that nj is the number of nodes at level j in tree t. With our previous definitions, a level number sequence starts with a 1 and consists of eventually null integers. Figure 1 displays a tree whose level number sequence is (1, 1, 3, 9, 25, 0, 0 , . . . ) , each node being labelled with its depth. (We are indebted to Ms C. Cabart for providing this computer generated diagram which is built after a natural botanical growth model for coffee and tree shrubs.) Let t be a binary tree as defined for instance in Knuth's book [7]: such trees are: (i) rooted, i.e., a certain node is distinguished as the root of the tree; (ii) binary, that is to say, each node has either 0 or 2 descendents; (iii) planar, i.e., subtrees han~ng from a binary node are distinguished as left or right subtrees. (Note planarity is not essential here). In an unpublished paper [2], Clowes, Mitrani and Wilson address the problem of determining the number of distinct level number sequences associated to all binary trees formed with n binary nodes. Without loss of generality, we shall define level number sequences for binary trees by taking here nj to be the number of binary nodes at depth j in the tree. (In computer science terms, this amount to considering nullary nodes as null pointers). Let Hn be the set of all such sequences, and let Hn = card(Hn). It is clear that an element of H,, is formed from a sequence (na, n2, • • •, nk) of non-zero integers followed by an infinite sequence of trailing zeroes: nj = 0 for j > k. The nj with