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 <IMG WIDTH=104 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66876" SRC="img1546.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1546.gif"  >.
Recall that two binomial trees of order <I>k</I> can be combined
to obtain a binomial tree of order <I>k</I>+1.
E.g.,  <IMG WIDTH=97 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66882" SRC="img1547.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1547.gif"  >.
But this is just like adding binary digits!
In binary notation, the sum 27+10 is calculated like this:
<DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=7 BORDER RULES=GROUPS>
<COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
	   </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	+</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<P>
	   </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>0</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>1</TD></TR>
</TBODY>
</TABLE>
<BR> 
</P></DIV>
The merging of  <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif"  > and  <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66890" SRC="img1549.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1549.gif"  > is done in the same way:
<DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=8 BORDER RULES=GROUPS>
<COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER>
<TBODY>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
	   </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>     </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66602" SRC="img1504.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1504.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66738" SRC="img1520.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1520.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66594" SRC="img1501.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1501.gif"  > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66888" SRC="img1548.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1548.gif"  > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 
	+</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>     </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>     </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66738" SRC="img1520.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1520.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66596" SRC="img1502.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1502.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66914" SRC="img1550.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1550.gif"  > </TD></TR>
<TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>
<P>
	   </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66916" SRC="img1551.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1551.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66922" SRC="img1552.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1552.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=6 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline63650" SRC="img1117.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1117.gif"  ></TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=18 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66594" SRC="img1501.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1501.gif"  > </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=22 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline66928" SRC="img1553.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1553.gif"  > </TD></TR>
</TBODY>
</TABLE>
</P></DIV>
Therefore, the result is  <IMG WIDTH=130 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66930" SRC="img1554.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1554.gif"  >.
<P>
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