| Summary: | 碩士 === 國立交通大學 === 應用數學系所 === 105 === The Horton-Strahler number, proposed by Robert E. Horton and Arthur N. Strahler, was originally introduced for the classification of river systems. However, it was later also used in Computer Science (where it is called the register function) and found many other applications in diverse areas.
One important parameter related to the Horton-Strahler number is the number of $r$-branches. For a complete binary trees that is chosen uniformly at random, stochastic properties of this number were analyzed in an old paper of John M. Moon with a recursive approach based on the root decomposition of a tree. Recently, the same parameter was also analyzed in a paper of Benjamin Hackl, Clemens Heuberger and Helmut Prodinger with a different approach, namely by letting the tree grow via replacing nodes by chains. The latter paper did not include a detailed comparison of the two approaches.
The main goal of this thesis is to describe the above to approaches for the sake of comparing them. For the first approach, this will be done by rephrasing it with the modern theory of singularity analysis which was not fully developed when Moon wrote his paper. For the second approach, we will mainly follow the presentation in the paper of Hackl, Heuberger and Prodinger. Moreover, using recent result of Stephan Wagner, we will prove a central limit theorem as well (using the first approach).
Apart from analyzing the above number, Hack and Prodinger (jointly with Sara Kropf) also used the second approach for discussing related problems. We will show that these results could be also alternatively derived with the first approach above.
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