数据结构——二叉树(Binary Trees)
非线性数据结构
树的密度=结点数/高度
二叉树类
#pragma once class stnode
{
public:
int nodeValue; // node data stnode *left, *right, *parent; // child pointers and pointer to the node's parent // constructor
stnode (const int item, stnode *lptr = NULL, stnode*rptr = NULL, stnode *pptr = NULL):
nodeValue(item), left(lptr), right(rptr), parent(pptr)
{}
}; class stree
{
public:
stree(); // constructor. initialize root to NULL and size to 0
~stree(); // destructor
bool insert(const int item);
void Output(); private:
stnode *root; // pointer to tree root
int treeSize; // number of elements in the tree
stnode *creatSTNode(const int item, stnode *lptr, stnode *rptr, stnode*pptr);
}; stnode * stree::creatSTNode (const int item, stnode *lptr, stnode *rptr, stnode *pptr)
{
stnode*newNode; // initialize the data and all pointers
newNode = new stnode (item, lptr, rptr, pptr); return newNode;
}
完全二叉树(complete tree):
所有非叶子节点有两个子结点或一个左子结点。按从左到右顺序建树。
包含n个元素的完全二叉树 h=(int)(log2(n))
遍历:
1、层次遍历
按层从左到右遍历。
//层序遍历二叉树
void stree::LevelByLevel(stnode *root)
{
std::queue<stnode*> q;//建队
q.push(root);//根节点入队
stnode *cur;
while(!q.empty())
{
cur=q.front(); //获得队列的首元素
q.pop(); //首元素出队
temp.Format("%d ",cur->nodeValue); //输出结点的值
str+=temp; if(cur->left!=NULL) //若结点的左子树不空
{
q.push(cur->left);
}
if(cur->right!=NULL)//若结点的右子树不空
{
q.push(cur->right);
}
}
}
2、中序遍历(LDR)
先访问左结点数据,直到左节点为空则访问中间(父结点)数据,再访问右子结点数据。
盗一张百度的图:
3、前序遍历(DLR)
先访问父结点数据,再访问左子结点,最后右子结点。到达即访问,根结点在遍历的第一个。
上图的前序遍历结果为:ABDGJEHCFI
4、后序遍历(LRD)
先访问左子结点数据,再访问右子结点,最后父结点。根结点在遍历的最后一个。
上图的前序遍历结果为:JGDHEBIFCA
树的递归
1、递归遍历叶子结点
void CountLeaf(tnode<T> *t,int &count)
{
if(t!=NULL)
{
if(t->left==NULL&&t->right==NULL)
count++;
CountLeaf(t->left,count);
CountLeaf(t->right,count);
}
}
2、树的高度
int depth(tnode<T> *t)
{
int depthleft,depthright,depthval;
if(t==NULL)
depthval=-;
else
{
depthleft=depth(t->left);
depthright=depth(t->right);
depthval=+(depthleft>depthright? depthleft:depthright);
}
return depthval;
}
3、删除整树
void deleteTree(tnode<T> *t)
{
if(t!=NULL)
{
deleteTree(t->left);
deleteTree(t->right);
delete t;
}
}
树形输出:
#include <iomanip> // for setw()
#include <strstream> // for format conversion
#include <string> // node data formatted as a string
#include <queue>
#include <utility> using namespace std; class tnodeShadow
{
public:
string nodeValueStr; // formatted node value
int level,column;
tnodeShadow *left, *right; tnodeShadow ()
{}
};
/*
tnodeShadow *buildShadowTree(AVLnode *t, int level, int& column);
void displayTree(int maxCharacters);
void deleteShadowTree(tnodeShadow *t);
*/
tnodeShadow *AVLtree::buildShadowTree(AVLnode *t, int level, int& column)
{
tnodeShadow *newNode = NULL;
char text[];
ostrstream ostr(text,); if (t != NULL)
{
newNode = new tnodeShadow; tnodeShadow *newLeft = buildShadowTree(t->left, level+, column);
newNode->left = newLeft; ostr << t->nodeValue << ends;
newNode->nodeValueStr = text;
newNode->level = level;
newNode->column = column; column++; tnodeShadow *newRight = buildShadowTree(t->right, level+, column);
newNode->right = newRight;
} return newNode;
} void AVLtree::displayTree(int maxCharacters)
{
string label;
int level = , column = ;
int colWidth = maxCharacters + ; int currLevel = , currCol = ; if (treeSize == )
return; tnodeShadow *shadowRoot = buildShadowTree(root, level, column); tnodeShadow *currNode; queue<tnodeShadow *> q; q.push(shadowRoot); while(!q.empty())
{
currNode = q.front();
q.pop(); if (currNode->level > currLevel)
{
currLevel = currNode->level;
currCol = ;
cout << endl;
} if(currNode->left != NULL)
q.push(currNode->left); if(currNode->right != NULL)
q.push(currNode->right); if (currNode->column > currCol)
{
cout << setw((currNode->column-currCol)*colWidth) << " ";
currCol = currNode->column;
}
cout << setw(colWidth) << currNode->nodeValueStr;
currCol++;
}
cout << endl; deleteShadowTree(shadowRoot);
} void AVLtree::deleteShadowTree(tnodeShadow *t)
{
if (t != NULL)
{
deleteShadowTree(t->left);
deleteShadowTree(t->right);
delete t;
}
}
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