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<b><font face="Arial, Helvetica, sans-serif" size="4" color="#000000">〖<font color="#0033CC">lamma5.4</font>〗Let T be a tree with n nodes created as a result of
union2. No node in T has level greater then [ <img src="IMAGE/log2n.gif" width="61" height="24"> ] + 1
<font color="#FF0000">
proof:
</font> <font color="#FF0000">n = 1, </font> lemma is clearly true
<font color="#FF0000"> i < = n - 1</font>, assume that it is true for all trees with i nodes
<font color="#FF0000"> i = n</font> , T is a tree with n noes created by union2
last union operation union( k, j )
<font color="#0033CC">m -</font>--- the number of nodes in tree j (1 <= m <= n/2 )
<font color="#0033CC"> n - m</font> ---- the number of nodes in k
if the max level of any node in T is same as k
then max level in T <= [ <img src="IMAGE/lognm.gif" width="91" height="24"> ] + 1 <= [ <img src="IMAGE/log2n.gif" width="61" height="24"> ] +1
if the max level of any node in T is one more than in j
then max level in T
<= [ <img src="IMAGE/log2m.gif" width="74" height="24">] + 2 <= [ <img src="IMAGE/log2n.gif" width="61" height="24">/2 ] +2 <= [ <img src="IMAGE/log2n.gif" width="61" height="24"> ] +1
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