闵老板の帖子: 日撸 Java 三百行（21-30天，树与二叉树）

1. 二叉树的深度遍历的递归实现

树是数据结构中的重中之重, 特别是二叉树.许多实际问题抽象出来的数据结构往往是二叉树形式, 即使是一般的树也能简单地转换为二叉树, 而且二叉树的存储结构及其算法都较为简单, 因此二叉树显得特别重要. 二叉树特点是每个结点最多只能有两棵子树, 且有左右之分.

这里用递归的方法为二叉树写了几个方法:

1. 遍历. 包括前序, 中序, 后序遍历.

2. 计算深度的方法.

3. 计算总结点数的方法.

package datastructure.tree;public class BinaryTree {// The value;char value;// The left child;BinaryTree leftChild;// The right child;BinaryTree rightChild;/*** ********************* Construct a tree-node.** @param paraValue The value you given.*********************/public BinaryTree(char paraValue) {value = paraValue;leftChild = null;rightChild = null;}// Of one constructor/*** ********************* Construct a tree with 7 nodes.** @return The tree.*********************/public static BinaryTree manualConstructTree() {// The root of the tree.BinaryTree resultTree = new BinaryTree('A');// The other nodes.BinaryTree tempTreeB = new BinaryTree('B');BinaryTree tempTreeC = new BinaryTree('C');BinaryTree tempTreeD = new BinaryTree('D');BinaryTree tempTreeE = new BinaryTree('E');BinaryTree tempTreeF = new BinaryTree('F');BinaryTree tempTreeG = new BinaryTree('G');// Link them.resultTree.leftChild = tempTreeB;resultTree.rightChild = tempTreeC;tempTreeB.rightChild = tempTreeD;tempTreeC.leftChild = tempTreeE;tempTreeD.leftChild = tempTreeF;tempTreeD.rightChild = tempTreeG;return resultTree;}// Of manualConstrucTree/*** ********************* Pre-Order.*********************/public void preOrder() {System.out.print(value + " ");if(leftChild != null) {leftChild.preOrder();}// Of ifif(rightChild != null) {rightChild.preOrder();}// Of if}// Of preOrder/*** ********************* In-Order.*********************/public void inOrder() {if(leftChild != null) {leftChild.inOrder();}// Of ifSystem.out.print(value + " ");if(rightChild != null) {rightChild.inOrder();}// Of if}// Of inOrder./*** ********************* Post-Order.*********************/public void postOrder() {if(leftChild != null) {leftChild.postOrder();}// Of ifif(rightChild != null) {rightChild.postOrder();}// Of ifSystem.out.print(value + " ");}// Of postOrder./*** ********************* Calculate the depth of the tree.** @return The depth.*********************/public int getDepth() {if(leftChild == null && rightChild == null) {return 1;}// Of if int tempLeftDepth = 0, tempRightDepth = 0;if(leftChild != null) {tempLeftDepth = leftChild.getDepth();}// Of ifif(rightChild != null) {tempRightDepth = rightChild.getDepth();}// Of ifif(tempLeftDepth >= tempRightDepth) {return tempLeftDepth+1;}// Of if else{return tempRightDepth+1;}// Of else}// Of getDepth/*** ********************* Calculate all nodes.** @return The nodes of the tree.*********************/public int getNumNodes() {if ((leftChild == null) && (rightChild == null)) {return 1;} // Of ifint tempLeftNodes = 0, tempRightNodes = 0;if (leftChild != null) {tempLeftNodes = leftChild.getNumNodes();} // Of ifif (rightChild != null) {tempRightNodes = rightChild.getNumNodes();} // Of ifreturn tempLeftNodes + tempRightNodes + 1;}// Of getNumNodes./************************ The entrance of the program.* * @param args Not used now.**********************/public static void main(String args[]) {BinaryTree tempTree = manualConstructTree();System.out.println("Pre-order visit:");tempTree.preOrder();System.out.println("\rIn-order visit:");tempTree.inOrder();System.out.println("\rPost-order visit:");tempTree.postOrder();System.out.println("\rThe depth is: " + tempTree.getDepth());System.out.println("The number of nodes is: " + tempTree.getNumNodes());}// Of main}// Of BinaryCharTree

输出:

Pre-order visit:
A B D F G C E
In-order visit:
B F D G A E C
Post-order visit:
F G D B E C A
The depth is: 4
The number of nodes is: 7

2. 二叉树的存储

我们可以完全满二叉树的角度广度优先遍历的角度来考虑这个问题: 每个节点都有一个 name 及其在二叉树中的位置. 令根节点的位置为 0; 则第 2 层节点的位置依次为 1 至 2; 第 3 层节点的位置依次为 3 至 6. 以此类推.

把昨天那个例子所对应的二叉树画出来, 我们有两种方法:

1.空使用 0 来表示, 可以用一个向量来存储:
        [A, B, C, 0, D, E, 0, 0, 0, F, G]
        优点: 仅需要一个向量, 简单直接.
        缺点: 对于实际的二叉树, 很多子树为空, 导致大量的 0 值.
    2.使用压缩存储方式, 即将节点的位置和值均存储. 可表示为两个向量:
        [0, 1, 2, 4, 5, 9, 10]
        [A, B, C, D, E,F, G]

我们用队列的方式来定义. 与之前不同的是, 这次定义用的是Object类型, 它可以接收任意类型的数据.

package datastructure;public class CircularObjectQueue {// The total space. But the actual sapce will be one less.public static final int MAX_SIZE = 10;// The data;Object[] data;// The index of the head.int front;// The index of the tail.int rear;/*** ********************* The constructor.*********************/public CircularObjectQueue() {data = new Object[MAX_SIZE];front = 0;rear = 0;}// Of the constructor/*** ********************* Enqueue.** @param paraValue The given value.* @return true or false.*********************/public boolean enqueue(Object paraValue) {if(front == (rear + 1) % MAX_SIZE) {// System.out.println("Queue full");return false;}data[rear % MAX_SIZE] = paraValue;rear++;return true;}// Of enqueue/*** ********************* Dequeue.** @return The dequeued value.*********************/public Object dequeue() {Object resultValue;if(front == rear) {// System.out.println("The queue is empty, dequeue fail");return null;}// Of ifresultValue = data[front % MAX_SIZE];front++;return resultValue;}// Of dequeue/*** ********************** Output the values of all member variables, and separated by comma.** @return A new string.**********************/public String toString() {String resultString = "";if(front == rear) {return "empty";}// Of iffor(int i = front; i < rear - 1; i++) {resultString += data[i % MAX_SIZE] + ", ";}// Of for i resultString += data[(rear-1) % MAX_SIZE];return resultString;}// Of toString
}// Of class CircularObjectQueue

下面是老师的代码实现. 将内容位置应在之前写的BinaryTree.java中.

需要注意的是,节点的索引从0开始. 那么节点的左孩子的索引是其索引的二倍+1, 右孩子是其索引的二倍+2.

 /*** The values of nodes according to breadth first traversal.*/char[] valuesArray;/*** The indices in the complete binary tree.*/int[] indicesArray;/*********************** Convert the tree to data arrays, including a char array and an int array.* The results are stored in two member variables.* * @see #valuesArray* @see #indicesArray**********************/public void toDataArrays() {//Initialize arrays.int tempLength = getNumNodes();valuesArray = new char[tempLength];indicesArray = new int[tempLength];int i = 0;//Traverse and convert at the same time.CircleObjectQueue tempQueue = new CircleObjectQueue();tempQueue.enqueue(this);CircleIntQueue tempIntQueue = new CircleIntQueue();tempIntQueue.enqueue(0);BinaryCharTree tempTree = (BinaryCharTree) tempQueue.dequeue();int tempIndex = tempIntQueue.dequeue();while (tempTree != null) {valuesArray[i] = tempTree.value;indicesArray[i] = tempIndex;i++;if (tempTree.leftChild != null) {tempQueue.enqueue(tempTree.leftChild);tempIntQueue.enqueue(tempIndex * 2 + 1);} // Of ifif (tempTree.rightChild != null) {tempQueue.enqueue(tempTree.rightChild);tempIntQueue.enqueue(tempIndex * 2 + 2);} // Of iftempTree = (BinaryCharTree) tempQueue.dequeue();tempIndex = tempIntQueue.dequeue();} // Of while}// Of toDataArrays/************************ The entrance of the program.* * @param args*            Not used now.**********************/public static void main(String args[]) {BinaryCharTree tempTree = manualConstructTree();System.out.println("\r\nPreorder visit:");tempTree.preOrderVisit();System.out.println("\r\nIn-order visit:");tempTree.inOrderVisit();System.out.println("\r\nPost-order visit:");tempTree.postOrderVisit();System.out.println("\r\n\r\nThe depth is: " + tempTree.getDepth());System.out.println("The number of nodes is: " + tempTree.getNumNodes());tempTree.toDataArrays();System.out.println("The values are: " + Arrays.toString(tempTree.valuesArray));System.out.println("The indices are: " + Arrays.toString(tempTree.indicesArray));}// Of main

输出:

Preorder visit:
A B D F G C E
In-order visit:
B F D G A E C
Post-order visit:
F G D B E C A
The depth is: 4
The number of nodes is: 7
The values are: [A, B, C, D, E, F, G]
The indices are: [0, 1, 2, 4, 5, 9, 10]

3. 使用具有通用性的队列

我们刚刚已经知道, Object可以用来接收任意类型数据, 那么我们在定义队列的时候, 也不再需要专门去为不同的数据类型创建不同的队列了.

我们可以直接这样定义数据:

Object[] data;

但是在使用的时候, 需要利用强制类型转换使其变成本身的类别. 比如

tempTree = (BinaryCharTree) tempQueue.dequeue();

4. 二叉树的建立

这里是二叉树的另一种构造方法, 通过输入相应的valuesArray、indicesArray来构造.

主要有三个步骤:

1. 创建一个Binary类型的数组来存储我们需要的所有节点.

2. 采用双重循环的方式判断是否为某个节点的左右孩子. 这里设计的十分巧妙, 从除根节点外的第一个节点出发, 判断其与之前的节点是否满足为其左右孩子的条件, 若满足条件, 则用之前创建好的Binary类型节点链接起来, 然后跳出循环, 判断下一个节点, 以此直到所有节点判断完毕.

3. 链接完毕, 回到根节点.

/************************ The second constructor. The parameters must be correct since no validity* check is undertaken.* * @param paraDataArray    The array for data.* @param paraIndicesArray The array for indices.**********************/public BinaryTree(char[] paraValuesArray, int[] paraIndicesArray) {// Step 1. Use a sequential list to store all nodes.int tempNumNodes = paraValuesArray.length;BinaryTree[] tempAllNodes = new BinaryTree[tempNumNodes];for (int i = 0; i < tempNumNodes; i++) {tempAllNodes[i] = new BinaryTree(paraValuesArray[i]);} // Of for i// Step 2. Link these nodes.for (int i = 1; i < tempNumNodes; i++) {for (int j = 0; j < i; j++) {System.out.println("indices " + paraIndicesArray[j] + " vs. " + paraIndicesArray[i]);if (paraIndicesArray[i] == paraIndicesArray[j] * 2 + 1) {tempAllNodes[j].leftChild = tempAllNodes[i];System.out.println("Linking " + j + " with " + i);break;} else if (paraIndicesArray[i] == paraIndicesArray[j] * 2 + 2) {tempAllNodes[j].rightChild = tempAllNodes[i];System.out.println("Linking " + j + " with " + i);break;} // Of if} // Of for j} // Of for i//Step 3. The root is the first node.value = tempAllNodes[0].value;leftChild = tempAllNodes[0].leftChild;rightChild = tempAllNodes[0].rightChild;}// Of the the third constructor

5. 二叉树深度遍历的栈实现 (中序)

5.1 构建通用栈

在之前写的stack.java中稍作修改, 并添加了.isEmpty()方法.

package datastructure.Stack;public class ObjectStack {// The max depth of the stack.public static final int MAX_DEPTH = 10;// The actual depth of the stack.int depth;// The data.Object[] data;/*** ********************** Initialize an empty stack.**********************/public ObjectStack() {depth = 0;data = new Object[MAX_DEPTH];}// Of the constructor/*** ********************** Push a value into the stack. If the stack is full, push fail.* * @param tempNode The given value.* @return true or false.**********************/public boolean push(Object paraObject) {if (depth == MAX_DEPTH) {System.out.println("The stack is full, push fail");return false;} // Of ifdata[depth] = paraObject;depth++;return true;}// Of push/*** ********************** Pop the value at the of top of stack. If the stack is empty, pop fail.** @return The value at the of top of stack.**********************/public Object pop() {if (depth == 0) {System.out.println("The stack is empty, pop fail");return '\0';} // Of ifObject resultValue = data[depth - 1];depth--;return resultValue;}// Of pop/*** ********************** Output the values of all member variables.** @return A new string.**********************/public String toString() {String resultString = "";if (depth == 0) {return "empty";} // Of iffor (int i = 0; i < depth; i++) {resultString += data[i];} // Of for ireturn resultString;}// Of toString/*** ********************* Determine whether the stack is empty. True for empty.** @return true or false.*********************/public boolean isEmpty() {if(depth == 0) {return true;}// Of ifreturn false;}// Of isEmpty}// Of class ObjectStack

5.2 中序遍历

简单来说, 入左, 左空出左,入右...

    /************************ In-order with stack.**********************/public void inOrderWithStack() {ObjectStack tempStack = new ObjectStack();BinaryTree tempNode = this;while (! tempStack.isEmpty() || tempNode != null) {if (tempNode != null) {tempStack.push(tempNode);tempNode = tempNode.leftChild;} else {tempNode = (BinaryTree) tempStack.pop();System.out.print(tempNode.value + " ");tempNode = tempNode.rightChild;} // Of if} // Of while}// Of inOrderWithStack

6. 小结

看不懂就反复看, 或者画图画出来, 多敲几遍, 然后自己琢磨.

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