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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H3><A NAME="SECTION0012211000000000000000">Complete <I>N</I>-ary Trees</A></H3>
<P>
The definition for complete binary trees can be easily extended
to trees with arbitrary fixed degree <IMG WIDTH=43 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline66294" SRC="img1450.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1450.gif" > as follows:
<P>
<BLOCKQUOTE> <b>Definition (Complete <I>N</I>-ary Tree)</b><A NAME="defncompletent"> </A>
A <em>complete <I>N</I>-ary tree</em><A NAME=24281> </A><A NAME=24335> </A>
of height <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline63700" SRC="img1122.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1122.gif" >,
is an <I>N</I>-ary tree <IMG WIDTH=171 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66306" SRC="img1451.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1451.gif" >
with the following properties.
<OL><LI> If <I>h</I>=0, <IMG WIDTH=43 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline66310" SRC="img1452.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1452.gif" > for all <I>i</I>, <IMG WIDTH=69 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline66314" SRC="img1453.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1453.gif" >.<LI> For <I>h</I><I>></I>0 there exists a <I>j</I>, <IMG WIDTH=71 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66320" SRC="img1454.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1454.gif" > such that
<OL><LI> <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 a perfect binary tree of height <I>h</I>-1
for all <IMG WIDTH=81 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline66326" SRC="img1455.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1455.gif" >;<LI> <IMG WIDTH=14 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline63482" SRC="img1102.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1102.gif" > is a complete binary tree of height <I>h</I>-1; and,<LI> <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 a perfect binary tree of height <I>h</I>-2
for all <I>i</I>:<I>j</I><I><</I><I>i</I><I><</I><I>N</I>.
</OL></OL></BLOCKQUOTE>
<P>
Note that while it is expressed in somewhat different terms,
the definition of a complete <I>N</I>-ary tree is consistent
with the definition of a binary tree for <I>N</I>=2.
Figure <A HREF="page356.html#figtree17" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page356.html#figtree17"><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> shows an example of a complete ternary (<I>N</I>=3) tree.
<P>
<P><A NAME="24580"> </A><A NAME="figtree17"> </A> <IMG WIDTH=575 HEIGHT=181 ALIGN=BOTTOM ALT="figure24290" SRC="img1456.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1456.gif" ><BR>
<STRONG>Figure:</STRONG> A Complete Ternary Tree<BR>
<P>
<P>
Informally, a complete tree is a tree in which all the levels
are full except for the bottom level
and the bottom level is filled from left to right.
For example in Figure <A HREF="page356.html#figtree17" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page356.html#figtree17"><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 first three levels are full.
The fourth level which comprises nodes 14-21
is partially full and has been filled from left to right.
<P>
The main advantage of using complete binary trees is that
they can be easily stored in an array.
Specifically, consider the nodes of a complete tree
numbered consecutively in <em>level-order</em><A NAME=24585> </A> as they are
in Figures <A HREF="page355.html#figtree16" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page355.html#figtree16"><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> and <A HREF="page356.html#figtree17" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page356.html#figtree17"><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>.
There is a simple formula that relates the number of a node
with the number of its parent
and the numbers of its children.
<P>
Consider the case of a complete binary tree.
The root node is node 1 and its children are nodes 2 and 3.
In general, the children of node <I>i</I> are 2<I>i</I> and 2<I>i</I>+1.
Conversely, the parent of node <I>i</I> is <IMG WIDTH=32 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline66352" SRC="img1457.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1457.gif" >.
Figure <A HREF="page356.html#figheap1" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page356.html#figheap1"><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> illustrates this idea by showing how
the complete binary tree shown in Figure <A HREF="page355.html#figtree16" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page355.html#figtree16"><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>
is mapped into an array.
When using this approach,
the pointers are no longer explicitly recorded.
<P>
<P><A NAME="24837"> </A><A NAME="figheap1"> </A> <IMG WIDTH=575 HEIGHT=201 ALIGN=BOTTOM ALT="figure24590" SRC="img1458.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1458.gif" ><BR>
<STRONG>Figure:</STRONG> Array Representation of a Complete Binary Tree<BR>
<P>
<P>
A remarkable characteristic of complete trees is that filling the bottom level
from left to right corresponds to adding elements at the end of the array!
Thus, a complete tree containing <I>n</I> nodes occupies the first
<I>n</I> consecutive array positions.
<P>
The array subscript calculations given above
can be easily generalized to complete <I>N</I>-ary trees.
Assuming that the root occupies position 1 of the array,
its <I>N</I> children occupy positions 2, 3, ..., <I>N</I>+1.
In general, the children of node <I>i</I> occupy positions
<P> <IMG WIDTH=432 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath66288" SRC="img1459.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img1459.gif" ><P>
and the parent of node <I>i</I> is found at
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