Question 12.2.2: Prove that a balanced tree with n > 1 vertices has heigh......

Algorithms and Programming: Problems and Solutions [1518855]

Prove that a balanced tree with n > 1 vertices has height at most C log n for some constant C that does not depend on n.

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By induction over n, we prove that $Φ_{n+2} ≥ a^n$ where a is the larger root of the quadratic equation $a² = 1 + a,$ that is, $a = (\sqrt{5} + 1)/2$ . (This number is usually called “the golden mean”.) It remains to apply the preceding problem.

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