| Summary: | In this Master's thesis, we consider the Horton-Strahler number $S sb{ rm n}$ for various distributions of random binary trees with n nodes. For random equiprobable binary trees with n nodes, we give a new probabilistic proof of the well-known result that E$S sb{n}$ = log$ sb4$ n + O(1), and show that for every x $>$ 0, bf P} { vert S sb{n} - log sb4 n vert ge x } le {D over 4 sp{x}}, or some constant $D>0.$ For random binary tries and Patricia trees constructed from n i.i.d. sequences of independent bits with probability $p in (0, 1),$ we use fill-up levels to show that lim limits sb{n to infty}} { bf P} left {S sb{n} 0.$ For random binary search trees, we present simple bounds on $S sb{n}$. Finally, we discuss the use of the Horton-Strahler number in drawing trees. We illustrate a known algorithm for drawing "realistic-looking" trees and present a new algorithm for drawing "nice-looking" trees, both with PostScript implementations.
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