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<b><font face="Arial, Helvetica, sans-serif" size="4" color="#FF0000">2) The property of Binary Trees
</font><font face="Arial, Helvetica, sans-serif" size="4" color="#000000">
〖<font color="#0033CC">Lemma5.1</font>〗Maximum number Of nodes:
    (1)  The Maximum number of nodes on level i of a binary 
           tree is  <img src="IMAGE/2i-1.gif" width="102" height="29" vspace="0" align="bottom">
 
    (2)  The Maximum number of nodes in a binary tree of  
           depth k is   <img src="IMAGE/2k-1.gif" width="114" height="29" align="bottom">
  
<font color="#FF0000">Proof:</font>
(1) The proof is by induction on i
    <i><font color="#0033CC">induction base: </font></i>
          i = 1(root)       max_num ( on level 1) =  <img src="IMAGE/2i-10.gif" width="136" height="23">
    <font color="#0033CC"><i>induction hypotheses</i>:</font>
         For all j, 1<= j < i,       max_num ( on level j ) =  <img src="IMAGE/2j-1.gif" width="45" height="30" align="bottom">
    <font color="#0033CC"><i>induction step</i></font><i><font face="Arial, Helvetica, sans-serif" size="4">:</font></i> 
         Since each node in a binary tree has a maximum degree of 2, 
          max_num (on level i) = max_num(on level i -1)*2= <img src="IMAGE/2i-2.gif" width="153" height="30">

(2) The maximum number of nodes in a binary tree of depth k is: </font></b></pre>
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