Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα s (j), which is of orderj 1/2. Thej-th ”s-hyperoscillation”α 1(j)−α s+1(j) is given by 1/2α 1(s)+O(j −1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s 1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.
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Kirschenhofer, P., Prodinger, H. On the average hyperoscillations of planted plane trees. Combinatorica 2, 177–186 (1982). https://doi.org/10.1007/BF02579316
AMS subject classification (1980)
- 05 C 05
- 68 E 10