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<b><font face="Arial, Helvetica, sans-serif" size="4" color="#000000">【<font color="#FF0000">theorem</font>】any decision tree that sorts n distinct elements has a
height of at least <img src="image/text7-t.gif" width="89" height="24" align="middle"> +1
【<font color="#FF0000">corollary</font>】any algorithm that sort by comparison only must have
a worst computing time of <img src="image/text7-s.gif" width="115" height="24" align="middle">
<font color="#FF0000">proof:</font>
for every decision tree with n! leaves there is a path of length
c<img src="image/nlogn.gif" width="79" height="24" align="middle"> , c a constant, By the theorem, there is path length
<img src="image/text7-t.gif" width="89" height="24" align="middle">
<i> n! = n(n-1)(n-2), ..., (3) (2) (1) >=</i> <img src="image/text7-u.gif" width="87" height="26" align="middle">
so, <img src="image/text7-t.gif" width="89" height="24" align="middle"> >= (<img src="image/nlogn.gif" width="79" height="24" align="middle">/2) = O( <img src="image/nlogn.gif" width="79" height="24" align="middle">)</font></b></pre>
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