Bounding Maximum Degree D(n)

Lemma 21.1

, $\;\;\;\;\;$where F_k is a Fibonacci number.

$$F_k = \left\{ \begin{array}{ll} k & {\rm if} \; k < 2 \\ F_{k-1} + F_{k-2} & {\rm if} \; k \geq 2 \end{array} \right.$$

Lemma 21.3

For a node x in a Fibonacci heap, where k = degree(x),

size(x) $\geq F_{k+2} \geq \phi^k$, where

size(x) = #nodes in subtree rooted at x

Corollary 21.4

D(n) = O(lgn)

By Lemma 21.3, n $\geq$ size(x) $\geq \phi^k$, where n = nodes in Fibonacci Heap and k = degree of any node x.

Then , and k = = O(lg n).