Question 12.2.1: Find the minimal and maximal number of vertices in a balance......

Algorithms and Programming: Problems and Solutions [1518855]

Find the minimal and maximal number of vertices in a balanced tree of height n.

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The maximal number of vertices is equal to $2^{n+1} – 1$ . If $m_n$ is the minimal number of vertices, then $m_{n+2} = 1 +m_n + m_{n+1}$ . An easy induction argument gives $m_n = Φ_{n+3} – 1$ (where $Φ_{n}$ is the n-th Fibonacci number: $Φ_{1}=1 ,$ and $Φ_{2}=1 ,$ and $Φ_{n+2}=Φ_{n}+Φ_{n+1}$ ).

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