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二叉树

Author
smallmartial
Published
2022/04/29
Updated
2020/04/21
Article Outline
TOC
Collection Outline

二叉树

1.为什么需要树这种数据结构

  • 数组存储方式的分析 优点：通过下标方式访问元素，速度快。对于有序数组，还可使用二分查找提高检索速度。 缺点：如果要检索具体某个值，或者插入值(按一定顺序)会整体移动，效率较低
  • 链式存储方式的分析 优点：在一定程度上对数组存储方式有优化(比如：插入一个数值节点，只需要将插入节点，链接到链表中即可， 删除效率也很好)。 缺点：在进行检索时，效率仍然较低，比如(检索某个值，需要从头节点开始遍历
  • 树存储方式的分析 能提高数据存储，读取的效率, 比如利用 二叉排序树(Binary Sort Tree)，既可以保证数据的检索速度，同时也可以保证数据的插入，删除，修改的速度。

2.树示意图

1560596381763

3.二叉树的概念

  • 树有很多种，每个节点最多只能有两个子节点的一种形式称为二叉树。
  • 二叉树的子节点分为左节点和右节点。

1560596464787

  • 如果该二叉树的所有叶子节点都在最后一层，并且结点总数= 2^n -1 , n 为层数，则我们称为满二叉树。
  • 如果该二叉树的所有叶子节点都在最后一层或者倒数第二层，而且最后一层的叶子节点在左边连续，倒数第二层的叶子节点在右边连续，我们称为完全二叉树。

1560596539939

3.1二叉树遍历

  • 前序遍历: 先输出父节点，再遍历左子树和右子树

  • 中序遍历: 先遍历左子树，再输出父节点，再遍历右子树

  • 后序遍历: 先遍历左子树，再遍历右子树，最后输出父节点

  • 小结: 看输出父节点的顺序，就确定是前序，中序还是后序

3.2代码实现

package cn.smallmartial.tree;

/**
 * @Author smallmartial
 * @Date 2019/6/15
 * @Email [email protected]
 */
public class BinaryTreeDemo {
    public static void main(String[] args) {
        //创建一个二叉树
        BinaryTree binaryTree = new BinaryTree();
        HeroNode root = new HeroNode(1, "doodou");
        HeroNode heroNode2 = new HeroNode(2, "smallmartial");
        HeroNode heroNode3 = new HeroNode(3, "张三");
        HeroNode heroNode4 = new HeroNode(4, "李四");

        //先手动创建二叉树
        root.setLeft(heroNode2);
        root.setRiht(heroNode3);
        heroNode3.setRiht(heroNode4);
        binaryTree.setRoot(root);
        System.out.println("前序遍历");
        binaryTree.preOrder();

        System.out.println("中序遍历");
        binaryTree.infixOrder();
        System.out.println("后续序遍历");
        binaryTree.postOrder();
    }
}
//创建二叉树
class BinaryTree{

    private HeroNode root;

    public void setRoot(HeroNode root){
        this.root = root;
    }
    //前序遍历
    public void preOrder(){
        if (this.root != null){
            this.root.proOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }

    //中序遍历
    public void infixOrder(){
        if (this.root != null){
            this.root.infixOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
    //后续遍历
    public void postOrder(){
        if (this.root != null){
            this.root.postOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
}

class HeroNode{
    private int no;
    private String name;
    private HeroNode left;
    private HeroNode right;

    public HeroNode(int no, String name) {
        this.no = no;
        this.name = name;
    }

    public int getNo() {
        return no;
    }

    public void setNo(int no) {
        this.no = no;
    }

    public String getName() {
        return name;
    }

    public void setName(String name) {
        this.name = name;
    }

    public HeroNode getLeft() {
        return left;
    }

    public void setLeft(HeroNode left) {
        this.left = left;
    }

    public HeroNode getRiht() {
        return right;
    }

    public void setRiht(HeroNode riht) {
        this.right = riht;
    }

    @Override
    public String toString() {
        return "HeroNode{" +
                "no=" + no +
                ", name='" + name + '\'' +
                ", left=" + left +
                ", riht=" + right +
                '}';
    }
    //前序遍历
    public void proOrder(){
        System.out.println(this);
        //递归向左子树前序遍历
        if (this.left != null){
            this.left.proOrder();
        }
        //递归向右子树前序遍历
        if (this.right != null){
            this.right.proOrder();
        }
        //中序遍历
    }

    //中序遍历
    public void infixOrder(){
        //递归向左子树中序遍历
        if (this.left != null){
            this.left.infixOrder();
        }
        //输出父节点
        System.out.println(this);
        //递归向右子树中序遍历
        if (this.right != null){
            this.right.infixOrder();
        }
    }

    //后序遍历
    public void  postOrder(){
        if (this.left != null){
            this.left.postOrder();
        }
        if (this.right != null){
            this.right.postOrder();
        }
        System.out.println(this);
    }

}

运行结果

1560601642234

3.3二叉树-查找指定节点

  • 要求： 请编写前序查找，中序查找和后序查找的方法。 并分别使用三种查找方式，查找 heroNO = 5 的节点 并分析各种查找方式，分别比较了多少次

  • 代码

    package cn.smallmartial.tree;

/**
 * @Author smallmartial
 * @Date 2019/6/15
 * @Email [email protected]
 */
public class BinaryTreeDemo {
    public static void main(String[] args) {
        //创建一个二叉树
        BinaryTree binaryTree = new BinaryTree();
        HeroNode root = new HeroNode(1, "doodou");
        HeroNode heroNode2 = new HeroNode(2, "smallmartial");
        HeroNode heroNode3 = new HeroNode(3, "张三");
        HeroNode heroNode4 = new HeroNode(4, "李四");

        //先手动创建二叉树
        root.setLeft(heroNode2);
        root.setRiht(heroNode3);
        heroNode3.setRiht(heroNode4);
        binaryTree.setRoot(root);

//
//        System.out.println("前序遍历");
//        binaryTree.preOrder();
//
//        System.out.println("中序遍历");
//        binaryTree.infixOrder();
//
//        System.out.println("后续序遍历");
//        binaryTree.postOrder();

        //前序遍历
        System.out.println("前序遍历方式");
        HeroNode resNode = binaryTree.preOrderSerch(4);
        if (resNode != null){
            System.out.println("找到了信息为no="+resNode.getNo()+"name="+resNode.getName());
        }else {
            System.out.println("没有找到 no ="+5);
        }
    }
}
//创建二叉树
class BinaryTree{

    private HeroNode root;

    public void setRoot(HeroNode root){
        this.root = root;
    }
    //前序遍历
    public void preOrder(){
        if (this.root != null){
            this.root.proOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }

    //中序遍历
    public void infixOrder(){
        if (this.root != null){
            this.root.infixOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
    //后续遍历
    public void postOrder(){
        if (this.root != null){
            this.root.postOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
    //前序遍历查找
    public HeroNode preOrderSerch(int no){
        if (root != null){
            return root.proOrderserch(no);
        }else {
            return null;
        }
    }

    //中序遍历
    public HeroNode infixOrderSearch(int no){
        if (root != null){
            return root.infixOrderSearch(no);
        }else {
            return null;
        }
    }
    //后续遍历
    public HeroNode postOrderSearch(int no ){
        if (root != null){
            return root.postOrderSerach(no);
        }else {
            return null;
        }
    }

}

class HeroNode{
    private int no;
    private String name;
    private HeroNode left;
    private HeroNode right;

    public HeroNode(int no, String name) {
        this.no = no;
        this.name = name;
    }

    public int getNo() {
        return no;
    }

    public void setNo(int no) {
        this.no = no;
    }

    public String getName() {
        return name;
    }

    public void setName(String name) {
        this.name = name;
    }

    public HeroNode getLeft() {
        return left;
    }

    public void setLeft(HeroNode left) {
        this.left = left;
    }

    public HeroNode getRiht() {
        return right;
    }

    public void setRiht(HeroNode riht) {
        this.right = riht;
    }

    @Override
    public String toString() {
        return "HeroNode{" +
                "no=" + no +
                ", name='" + name + '\'' +
                ", left=" + left +
                ", riht=" + right +
                '}';
    }
    //前序遍历
    public void proOrder(){
        System.out.println(this);
        //递归向左子树前序遍历
        if (this.left != null){
            this.left.proOrder();
        }
        //递归向右子树前序遍历
        if (this.right != null){
            this.right.proOrder();
        }
        //中序遍历
    }

    //中序遍历
    public void infixOrder(){
        //递归向左子树中序遍历
        if (this.left != null){
            this.left.infixOrder();
        }
        //输出父节点
        System.out.println(this);
        //递归向右子树中序遍历
        if (this.right != null){
            this.right.infixOrder();
        }
    }

    //后序遍历
    public void  postOrder(){
        if (this.left != null){
            this.left.postOrder();
        }
        if (this.right != null){
            this.right.postOrder();
        }
        System.out.println(this);
    }

    //前序遍历查找
    public HeroNode proOrderserch(int no){
        if (this.no == no){
            return this;
        }

        HeroNode resNode = null;
        if (this.left != null){
            resNode = this.left.proOrderserch(no);
        }

        if (resNode != null){
            return resNode;
        }

        if (this.right != null){
            resNode = this.right.proOrderserch(no);
        }
        return resNode;
    }

    //中序遍历查找
    public  HeroNode infixOrderSearch(int no){

        HeroNode resNode = null;
        if (this.left != null){
            resNode = this.left.infixOrderSearch(no);
        }

        if (resNode != null){
            return resNode;
        }

        if (this.no == no){
            return this;
        }

        if (this.right != null){
            resNode = this.right.infixOrderSearch(no);
        }
        return resNode;
    }

    //后序遍历
    public HeroNode postOrderSerach(int no ){
        HeroNode resNode = null;
        if (this.left != null){
            resNode = this.left.infixOrderSearch(no);
        }

        if (resNode != null){
            return resNode;
        }
        if (this.right != null){
            resNode = this.right.infixOrderSearch(no);
        }

        if (this.no == no){
            return this;
        }
        return resNode;

    }

}

3.4二叉树的删除

  • 如果删除的节点是叶子节点，则删除该节点 如果删除的节点是非叶子节点，则删除该子树. 测试，删除掉 5号叶子节点 和 3号子树.

  • 代码

    package cn.smallmartial.tree;

/**
 * @Author smallmartial
 * @Date 2019/6/15
 * @Email [email protected]
 */
public class BinaryTreeDemo {
    public static void main(String[] args) {
        //创建一个二叉树
        BinaryTree binaryTree = new BinaryTree();
        HeroNode root = new HeroNode(1, "doodou");
        HeroNode heroNode2 = new HeroNode(2, "smallmartial");
        HeroNode heroNode3 = new HeroNode(3, "张三");
        HeroNode heroNode4 = new HeroNode(4, "李四");

        //先手动创建二叉树
        root.setLeft(heroNode2);
        root.setRiht(heroNode3);
        heroNode3.setRiht(heroNode4);
        binaryTree.setRoot(root);

//
//        System.out.println("前序遍历");
//        binaryTree.preOrder();
//
//        System.out.println("中序遍历");
//        binaryTree.infixOrder();
//
//        System.out.println("后续序遍历");
//        binaryTree.postOrder();

//        //前序遍历
//        System.out.println("前序遍历方式");
//        HeroNode resNode = binaryTree.preOrderSerch(4);
//        if (resNode != null){
//            System.out.println("找到了信息为no="+resNode.getNo()+"name="+resNode.getName());
//        }else {
//            System.out.println("没有找到 no ="+5);
//        }

        //测试删除

        System.out.println("删除前");
        binaryTree.preOrder();
        binaryTree.delNode(4);
        System.out.println("删除后");
        binaryTree.preOrder();
    }
}
//创建二叉树
class BinaryTree{

    private HeroNode root;

    public void setRoot(HeroNode root){
        this.root = root;
    }

    public void delNode(int no){
        if (root != null){
            if (root.getNo() == no){
                root = null;
            }else {
                root.delNode(no);
            }
        }
    }
    //前序遍历
    public void preOrder(){
        if (this.root != null){
            this.root.proOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }

    //中序遍历
    public void infixOrder(){
        if (this.root != null){
            this.root.infixOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
    //后续遍历
    public void postOrder(){
        if (this.root != null){
            this.root.postOrder();
        }else {
            System.out.println("二叉树为空，无法遍历");
        }
    }
    //前序遍历查找
    public HeroNode preOrderSerch(int no){
        if (root != null){
            return root.proOrderserch(no);
        }else {
            return null;
        }
    }

    //中序遍历
    public HeroNode infixOrderSearch(int no){
        if (root != null){
            return root.infixOrderSearch(no);
        }else {
            return null;
        }
    }
    //后续遍历
    public HeroNode postOrderSearch(int no ){
        if (root != null){
            return root.postOrderSerach(no);
        }else {
            return null;
        }
    }

}

class HeroNode{
    private int no;
    private String name;
    private HeroNode left;
    private HeroNode right;

    public HeroNode(int no, String name) {
        this.no = no;
        this.name = name;
    }

    public int getNo() {
        return no;
    }

    public void setNo(int no) {
        this.no = no;
    }

    public String getName() {
        return name;
    }

    public void setName(String name) {
        this.name = name;
    }

    public HeroNode getLeft() {
        return left;
    }

    public void setLeft(HeroNode left) {
        this.left = left;
    }

    public HeroNode getRiht() {
        return right;
    }

    public void setRiht(HeroNode riht) {
        this.right = riht;
    }

    @Override
    public String toString() {
        return "HeroNode{" +
                "no=" + no +
                ", name='" + name + '\'' +
                ", left=" + left +
                ", riht=" + right +
                '}';
    }

    //递归删除节点
    public void delNode(int no){

        if (this.left != null &&this.left.no == no){
            this.left = null;
            return;
        }

        if (this.right != null && this.right.no == no){
            this.right = null;
            return;
        }
        //向左子树递归删除
        if (this.left != null){
            this.left.delNode(no);
        }

        //向右递归删除
        if (this.right != null){
            this.right.delNode(no);
        }


  }
  //前序遍历
  public void proOrder(){
      System.out.println(this);
      //递归向左子树前序遍历
      if (this.left != null){
          this.left.proOrder();
      }
      //递归向右子树前序遍历
      if (this.right != null){
          this.right.proOrder();
      }
      //中序遍历
  }

  //中序遍历
  public void infixOrder(){
      //递归向左子树中序遍历
      if (this.left != null){
          this.left.infixOrder();
      }
      //输出父节点
      System.out.println(this);
      //递归向右子树中序遍历
      if (this.right != null){
          this.right.infixOrder();
      }
  }

  //后序遍历
  public void  postOrder(){
      if (this.left != null){
          this.left.postOrder();
      }
      if (this.right != null){
          this.right.postOrder();
      }
      System.out.println(this);
  }

  //前序遍历查找
  public HeroNode proOrderserch(int no){
      if (this.no == no){
          return this;
      }

      HeroNode resNode = null;
      if (this.left != null){
          resNode = this.left.proOrderserch(no);
      }

      if (resNode != null){
          return resNode;
      }

      if (this.right != null){
          resNode = this.right.proOrderserch(no);
      }
      return resNode;
  }

  //中序遍历查找
  public  HeroNode infixOrderSearch(int no){

      HeroNode resNode = null;
      if (this.left != null){
          resNode = this.left.infixOrderSearch(no);
      }

      if (resNode != null){
          return resNode;
      }

      if (this.no == no){
          return this;
      }

      if (this.right != null){
          resNode = this.right.infixOrderSearch(no);
      }
      return resNode;
  }

  //后序遍历
  public HeroNode postOrderSerach(int no ){
      HeroNode resNode = null;
      if (this.left != null){
          resNode = this.left.infixOrderSearch(no);
      }

      if (resNode != null){
          return resNode;
      }
      if (this.right != null){
          resNode = this.right.infixOrderSearch(no);
      }

      if (this.no == no){
          return this;
      }
      return resNode;

  }

}


- 运行结果

  ![1560604802227](assets/1560604802227.png)
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