\documentclass[UTF8]{article}
\usepackage{tikz}
\usepackage{CTEX}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{xcolor}
\begin{document}
\title{Algorithm homework 2}
\author{****\\
21617019}
\date{2016-10-17}
\maketitle
\section*{6.3-3}
\begin{flushleft}
证明：\\
$\because$\quad 下标为 i 的父节点的子节点为 2i 和 2i+1 ,所以节点为 n 的叶节点的父节点应该为$\lfloor n/2 \rfloor$,叶节点的个数为$Num=n-\lfloor n/2\rfloor=\lceil n/2\rceil$。\\
高度为h 的节点个数$Num_h$满足:
\begin{align*}
&Num_0=\lceil n/2\rceil\\
&Num_{h+1}=\lceil Num_h/2\rceil
\end{align*}
$\therefore\quad$距离叶节点距离为h，其个数为 $Num/2^h=\lceil n/2^{h+1}\rceil$
\end{flushleft}
\section*{6.4-1}
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box/.style={circle,
minimum width=5pt, inner sep=3pt,
draw=gray,thick, fill=white}
}
\begin{center}
%aaaaaaaaaaaaaaaaaaaaaa
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a.最大堆
\end{center}
%bbbbbbbbbbbbbbbbbbbbbbbbbbb
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child {node[box] {8}[sibling distance=40pt]
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}
}
child {node[box] {7}}
}
child {node[box] {17}[sibling distance=40pt]
child {node[box] {4}}
child {node[box] {2}}
};
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] {25};
\end{tikzpicture}\\
(b)
\end{minipage}
%ccccccccccccccccccccccccccc
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {17}[sibling distance=80pt]
child {node[box] {13}[sibling distance=40pt]
child {node[box] {8}[sibling distance=40pt]
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child {node[box] {5}[sibling distance=40pt]
child {node[box] {4}}
child {node[box] {2}}
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\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(4,5){25};
\node[box] (a) at(3,5){20};
\end{tikzpicture}\\
(c)
\end{minipage}
\\%%%%%%%%ddddddddddddddddddddddd
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {13}[sibling distance=80pt]
child {node[box] {8}[sibling distance=40pt]
child {node[box] {2}[sibling distance=40pt]
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child {node[box] {5}[sibling distance=40pt]
child {node[box] {4}}
};
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(4,5){25};
\node[box] (a) at(3,5){20};
\node[box] (a) at(2,5){17};
\end{tikzpicture}\\
(d)
\end{minipage}
%%%%%%%eeeeeeeeeeeeeeeeee
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {8}[sibling distance=80pt]
child {node[box] {7}[sibling distance=40pt]
child {node[box] {2}[sibling distance=40pt]
}
child {node[box] {4}}
}
child {node[box] {5}[sibling distance=40pt]
};
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(4,5){25};
\node[box] (a) at(3,5){20};
\node[box] (a) at(2,5){17};
\node[box] (a) at(1,5){13};
\end{tikzpicture}\\
(e)
\end{minipage}
\\%%%%%%%%ffffffffffffffffffffffffff
\begin{minipage}[t]{0.6\textwidth}
\centering
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child {node[box] {4}[sibling distance=40pt]
child {node[box] {2}[sibling distance=40pt]
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child {node[box] {5}[sibling distance=40pt]
};
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(4,5){25};
\node[box] (a) at(3,5){20};
\node[box] (a) at(2,5){17};
\node[box] (a) at(1,5){13};
\node[box] (a) at(0,5){8};
\end{tikzpicture}\\
(f)
\end{minipage}
%%%%ggggggggggggggggggggggggggggggg
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {5}[sibling distance=80pt]
child {node[box] {4}[sibling distance=40pt]
}
child {node[box] {2}[sibling distance=40pt]
};
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(5,5){25};
\node[box] (a) at(4,5){20};
\node[box] (a) at(3,5){17};
\node[box] (a) at(2,5){13};
\node[box] (a) at(1,5){8};
\node[box] (a) at(0,5){7};
\end{tikzpicture}\\
(g)
\end{minipage}
\\%%%%%%%%hhhhhhhhhhhhhhhhhhhhhhhhhhh
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {4}[sibling distance=80pt]
child {node[box] {2}[sibling distance=40pt]
}
;
\end{tikzpicture}\\
\begin{tikzpicture}
\node[box] (b) at(5,5){25};
\node[box] (a) at(4.5,5){20};
\node[box] (a) at(4,5){17};
\node[box] (a) at(3.5,5){13};
\node[box] (a) at(3,5){8};
\node[box] (a) at(2.5,5){7};
\node[box] (a) at(2,5){5};
\end{tikzpicture}\\
(h)
\end{minipage}
%%%%%%%iiiiiiiiiiiiiiiiiiiiiii
\begin{minipage}[t]{0.6\textwidth}
\centering
\begin{tikzpicture}
\node[box] {2};
\end{tikzpicture}\\
\begin{tikzpicture}
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\node[box] (a) at(0,5){4};
\end{tikzpicture}\\
(i)
\end{minipage}
\\\\\\
排序结果：\\
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\node[box] (a) at(1,5){2};
\end{tikzpicture}\\
\end{document}
