Depth and path length of heap ordered trees

Abstract. A heap ordered tree of size $n$ is a planted plane tree together with a bijection from the nodes to the set $\{1,\dots,n\}$ which is monotonically increasing when going from the root to the leaves. In a recent paper by Chen and Ni, the expectation and the variance of the depth of a random node in a random heap ordered tree of size $n$ was considered. Here, we give additional results concerning level polynomials. From a historical point of view, we trace these trees back to an earlier paper by Prodinger and Urbanek and find formulae that are proved in the paper by Chen and Ni by ad hoc computations already in a classic book by Greene and Knuth. Also, we mention that a paper by Bergeron, Flajolet and Salvy develops a more general theory of increasing trees, based on simply generated families of trees. Furthermore we consider the path length which is a natural concept when dealing with the depth. Expectation and variance are obtained, both explicitly and asymptotically.

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Depth of nodes

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