The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees

Abstract : We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.
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Communication dans un congrès
Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.527-548, 2010, DMTCS Proceedings
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Götz Olaf Munsonius. The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees. Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.527-548, 2010, DMTCS Proceedings. 〈hal-01185577〉

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