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<b><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"> 4. <font color="#FF0000">Number of distinct binary trees</font>
    to obtain the  number of distinct binary tree with n nodes,
    we must solve the recurrence of Eq.(1). we let    
                                 <img src="IMAGE/text5-z19.gif" width="102" height="39">                                                        <font color="#FF0000">Eq.(1).</font>
          x<img src="IMAGE/b2.gif" width="20" height="27" align="bottom">(x)= B(x) -1                                                            

  using the formula to solve quadratics and the fact 
  that b(0) = b<font size="2"> 0 </font>= 1 we get:                                          
                             <img src="IMAGE/text5-z17.gif" width="139" height="46">                                                     <font color="#FF0000"> Eq.(2)</font>

  using the binomial theorem to expand <img src="IMAGE/text5-z20.gif" width="102" height="26" align="middle">  to obtain:
 <img src="IMAGE/text5-z21.gif" width="642" height="67" align="middle">      <font color="#FF0000">Eq.(3)</font>

  comparing Eq.(2) and Eq.(3) we see that bn, which is the coefficient
   of <img src="IMAGE/text5-z22.gif" width="23" height="17" align="middle">   in B(x), is:  <img src="IMAGE/text5-z23.gif" width="176" height="66" align="middle">
                                <img src="IMAGE/text5-z24.gif" width="163" height="69" align="middle">                  <img src="IMAGE/text5-z25.gif" width="172" height="27">

   </font><font face="Arial, Helvetica, sans-serif" size="4" color="#FF0000">exercises page 235/1   page 254/1</font></b>


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