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<TITLE>Removing Items from an M-Way Search Tree</TITLE>
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<b>Data Structures and Algorithms 
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION0011640000000000000000">Removing Items from an <I>M</I>-Way Search Tree</A></H2>
<P>
The procedure for removing items from an <I>M</I>-way search tree
follows directly from the algorithm for removing items from
a binary search tree given in Section&nbsp;<A HREF="page318.html#secsrchtreebstout" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page318.html#secsrchtreebstout"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>.
The basic idea is that the item to be deleted is pushed down the tree
from its initial position to a node from which it can be easily deleted.
Clearly, items are easily deleted from leaf nodes.
In addition, consider an internal node of an <I>M</I>-way search tree of the form
<P> <IMG WIDTH=419 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath65574" SRC="img1381.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1381.gif"  ><P>
If both  <IMG WIDTH=30 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline64404" SRC="img1227.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1227.gif"  > and  <IMG WIDTH=13 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline63460" SRC="img1099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1099.gif"  > are empty trees,
then the key  <IMG WIDTH=10 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64406" SRC="img1228.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1228.gif"  > can be deleted from <I>T</I>
by removing both  <IMG WIDTH=10 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64406" SRC="img1228.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1228.gif"  > and  <IMG WIDTH=13 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline63460" SRC="img1099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1099.gif"  >, say.
If  <IMG WIDTH=30 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline64404" SRC="img1227.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1227.gif"  > is non-empty,  <IMG WIDTH=10 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64406" SRC="img1228.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1228.gif"  > can be pushed down the tree by
swapping it with the largest key in  <IMG WIDTH=30 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline64404" SRC="img1227.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1227.gif"  >;
and if  <IMG WIDTH=13 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline63460" SRC="img1099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1099.gif"  > is non-empty,  <IMG WIDTH=10 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline64406" SRC="img1228.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1228.gif"  > can be pushed down the tree by
swapping it with the smallest key in  <IMG WIDTH=13 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline63460" SRC="img1099.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1099.gif"  >.
<P>
Program&nbsp;<A HREF="page339.html#progmwaytree5c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page339.html#progmwaytree5c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> gives the code for the <tt>Withdraw</tt>
function of the <tt>MWayTree</tt> class.
The general form of the algorithm follows that of the <tt>Withdraw</tt>
routine for the <tt>BST</tt> class (Program&nbsp;<A HREF="page319.html#progbst4c" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page319.html#progbst4c"><IMG  ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>).
<P>
<P><A NAME="21729">&#160;</A><A NAME="progmwaytree5c">&#160;</A> <IMG WIDTH=575 HEIGHT=678 ALIGN=BOTTOM ALT="program21665" SRC="img1382.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1382.gif"  ><BR>
<STRONG>Program:</STRONG> <tt>MWayTree</tt> Class <tt>Withdraw</tt> Member Function Definition<BR>
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