二叉树基本知识

Posted 2021-04-09 Android部落格

1 基本概念

• 节点的度

节点拥有的子树的个数称为该节点的度。

• 树的度

树中所有节点的度的最大值称为该树的度。

• 节点的层次

从根节点到树中某个节点所经路径上的分支数称为该节点的层次。

• 树的深度

树中所有节点的层次的最大值称为该树的深度。

上图中除最底层的叶子结点，所有节点的度为2，树的度为2，节点HIJ的层次为3，所以树的深度为3。

性质1：

若根节点的层次为0， 则非空二叉树第i层上最多有2^i个节点。

性质2：

若二叉树根节点的深度为0，则深度为K的二叉树最大节点数为2^(k + 1) - 1个。

性质3：

对于非空二叉树，如果叶子结点的个数为n，度为2的节点数为m，则有：n = m + 1。

性质4：

对于具有n个节点的完全二叉树，如果按照从上往下从左往右的顺序对所有节点从0开始编号，则有：

如果2i + 1 < n，那么i节点的左孩子节点序号为2i + 1，否则节点i没有左节点。

如果2i + 2 < n，那么i节点的右孩子节点序号为2i + 2，否则节点i没有右节点。

二叉树的数据一般用数组或链表存储。

2 创建二叉树

首先定义二叉树的基本数据结构：

public class BinaryTreeBase { public BinaryTreeBase lChild; public BinaryTreeBase rChild; public int value;
 public BinaryTreeBase(int value) { this.value = value; }}

接着创建二叉树：

private static BinaryTreeBase buildBinaryTree(BinaryTreeBase binaryTreeBase, int index) { if (index >= size / 2) { return null; } if (binaryTreeBase == null) { binaryTreeBase = new BinaryTreeBase(number[index]); } int leftIndex = 2 * index + 1; int rightIndex = 2 * index + 2; if (leftIndex < size) { binaryTreeBase.lChild = new BinaryTreeBase(number[leftIndex]); buildBinaryTree(binaryTreeBase.lChild, leftIndex); } if (rightIndex < size) { binaryTreeBase.rChild = new BinaryTreeBase(number[rightIndex]); buildBinaryTree(binaryTreeBase.rChild, rightIndex); } return binaryTreeBase;}
private static int[] number = {15, 28, 33, 16, 49, 10, 34, 61, 27, 36, 57, 68, 71, 16, 87, 90, 134, 243, 8, 51, 57};private static int size = number.length;private static BinaryTreeBase binaryTreeBase = null;
public static void start() { binaryTreeBase = buildBinaryTree(binaryTreeBase, 0);}

3 二叉树前序遍历

/* *@Params :root,left,right *@Author :chuck-os *@Date :2020/7/7 */public static void frontScan(BinaryTreeBase binaryTreeBase1) { if (binaryTreeBase == null) { binaryTreeBase = buildBinaryTree(binaryTreeBase, 0); } if (binaryTreeBase1 == null) { return; } Log.d(TAG, "frontScan number is:" + binaryTreeBase1.value); frontScan(binaryTreeBase1.lChild); frontScan(binaryTreeBase1.rChild);}

4 二叉树中序遍历

/* *@Params :left,root,right *@Author :chuck-os *@Date :2020/7/7 */public static void middleScan(BinaryTreeBase binaryTreeBase1) { if (binaryTreeBase == null) { binaryTreeBase = buildBinaryTree(binaryTreeBase, 0); } if (binaryTreeBase1 == null) { return; } middleScan(binaryTreeBase1.lChild); Log.d(TAG, "middleScan number is:" + binaryTreeBase1.value); middleScan(binaryTreeBase1.rChild);}

5 二叉树后序遍历

/* *@Params :left,right,root *@Author :chuck-os *@Date :2020/7/7 */public static void backScan(BinaryTreeBase binaryTreeBase1) { if (binaryTreeBase == null) { binaryTreeBase = buildBinaryTree(binaryTreeBase, 0); } if (binaryTreeBase1 == null) { return; } backScan(binaryTreeBase1.lChild); backScan(binaryTreeBase1.rChild); Log.d(TAG, "backScan number is:" + binaryTreeBase1.value);}

6 二叉树层序遍历

/* *@Params :from one layer to another layer *@Author :chuck-os *@Date :2020/7/7 */public static void layerScan() { if (binaryTreeBase == null) { binaryTreeBase = buildBinaryTree(binaryTreeBase, 0); } Queue<BinaryTreeBase> queue = new LinkedList(); BinaryTreeBase tempBinaryTree = binaryTreeBase; queue.add(tempBinaryTree); while (tempBinaryTree != null) { tempBinaryTree = queue.poll(); if (tempBinaryTree == null) { break; } Log.d(TAG, "layerScan number is:" + tempBinaryTree.value); if (tempBinaryTree.lChild != null) { queue.add(tempBinaryTree.lChild); } if (tempBinaryTree.rChild != null) { queue.add(tempBinaryTree.rChild); } }}

7 使用栈实现后序遍历

public static void backScanWithStack(){ if(binaryTreeBase == null){ binaryTreeBase = buildBinaryTree(binaryTreeBase, 0); } Stack<BinaryTreeBase> stack = new Stack<BinaryTreeBase>(); BinaryTreeBase tempBinaryTree = binaryTreeBase; stack.push(tempBinaryTree);  BinaryTreeBase lastVisitBinaryTree = null;  while(tempBinaryTree != null){ if(tempBinaryTree.lChild != null){ System.out.println("backScanWithStack left number is:" + tempBinaryTree.lChild.value); stack.push(tempBinaryTree.lChild); } tempBinaryTree = tempBinaryTree.lChild; }
 while(!stack.isEmpty()){ tempBinaryTree = stack.pop(); BinaryTreeBase rightChild = tempBinaryTree.rChild; if(rightChild != null && rightChild != lastVisitBinaryTree){ stack.push(tempBinaryTree); tempBinaryTree = rightChild; while(tempBinaryTree != null){ stack.push(tempBinaryTree); tempBinaryTree = tempBinaryTree.lChild; } } else { System.out.println("backScanWithStack number is:" + tempBinaryTree.value); lastVisitBinaryTree = tempBinaryTree; } }}

后序遍历示意图如下：