二叉樹(BinaryTree)是n(n≥0)個結點的有限集,它或者是空集(n=0),或者由一個根結點及兩棵互不相交的、分別稱作這個根的左子樹和右子樹的二叉樹組成。
這個定義是遞歸的。由於左、右子樹也是二叉樹, 因此子樹也可爲空樹。下圖中展現了五種不同基本形態的二叉樹。
其中 (a) 爲空樹, (b) 爲僅有一個結點的二叉樹, (c) 是僅有左子樹而右子樹爲空的二叉樹, (d) 是僅有右子樹而左子樹爲空的二叉樹, (e) 是左、右子樹均非空的二叉樹。這裏應特別注意的是,二叉樹的左子樹和右子樹是嚴格區分並且不能隨意顛倒的,圖 (c) 與圖 (d) 就是兩棵不同的二叉樹。
二叉樹的遍歷
對於二叉樹來講最主要、最基本的運算是遍歷。
遍歷二叉樹 是指以一定的次序訪問二叉樹中的每個結點。所謂 訪問結點 是指對結點進行各種操作的簡稱。例如,查詢結點數據域的內容,或輸出它的值,或找出結點位置,或是執行對結點的其他操作。遍歷二叉樹的過程實質是把二叉樹的結點進行線性排列的過程。假設遍歷二叉樹時訪問結點的操作就是輸出結點數據域的值,那麼遍歷的結果得到一個線性序列。
從二叉樹的遞歸定義可知,一棵非空的二叉樹由根結點及左、右子樹這三個基本部分組成。因此,在任一給定結點上,可以按某種次序執行三個操作:
(1)訪問結點本身(N),
(2)遍歷該結點的左子樹(L),
(3)遍歷該結點的右子樹(R)。
以上三種操作有六種執行次序:
NLR、LNR、LRN、NRL、RNL、RLN。
注意:
前三種次序與後三種次序對稱,故只討論先左後右的前三種次序。
由於被訪問的結點必是某子樹的根,所以N(Node)、L(Left subtlee)和R(Right subtree)又可解釋爲根、根的左子樹和根的右子樹。NLR、LNR和LRN分別又稱爲先根遍歷、中根遍歷和後根遍歷。
二叉樹的java實現
首先創建一棵二叉樹如下圖,然後對這顆二叉樹進行遍歷操作(遍歷操作的實現分爲遞歸實現和非遞歸實現),同時還提供一些方法如獲取雙親結點、獲取左孩子、右孩子等。
java實現代碼:
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package study_02.datastructure.tree;
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import java.util.Stack;
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public class BinaryTree {
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private TreeNode root=null;
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public BinaryTree(){
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root=new TreeNode(1,"rootNode(A)");
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}
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public void createBinTree(TreeNode root){
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TreeNode newNodeB = new TreeNode(2,"B");
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TreeNode newNodeC = new TreeNode(3,"C");
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TreeNode newNodeD = new TreeNode(4,"D");
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TreeNode newNodeE = new TreeNode(5,"E");
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TreeNode newNodeF = new TreeNode(6,"F");
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root.leftChild=newNodeB;
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root.rightChild=newNodeC;
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root.leftChild.leftChild=newNodeD;
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root.leftChild.rightChild=newNodeE;
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root.rightChild.rightChild=newNodeF;
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}
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public boolean isEmpty(){
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return root==null;
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}
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public int height(){
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return height(root);
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}
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public int size(){
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return size(root);
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}
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private int height(TreeNode subTree){
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if(subTree==null)
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return 0;
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else{
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int i=height(subTree.leftChild);
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int j=height(subTree.rightChild);
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return (i<j)?(j+1):(i+1);
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}
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}
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private int size(TreeNode subTree){
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if(subTree==null){
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return 0;
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}else{
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return 1+size(subTree.leftChild)
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+size(subTree.rightChild);
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}
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}
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public TreeNode parent(TreeNode element){
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return (root==null|| root==element)?null:parent(root, element);
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}
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public TreeNode parent(TreeNode subTree,TreeNode element){
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if(subTree==null)
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return null;
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if(subTree.leftChild==element||subTree.rightChild==element)
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return subTree;
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TreeNode p;
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if((p=parent(subTree.leftChild, element))!=null)
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return p;
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else
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return parent(subTree.rightChild, element);
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}
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public TreeNode getLeftChildNode(TreeNode element){
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return (element!=null)?element.leftChild:null;
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}
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public TreeNode getRightChildNode(TreeNode element){
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return (element!=null)?element.rightChild:null;
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}
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public TreeNode getRoot(){
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return root;
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}
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public void destroy(TreeNode subTree){
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if(subTree!=null){
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destroy(subTree.leftChild);
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destroy(subTree.rightChild);
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subTree=null;
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}
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}
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public void traverse(TreeNode subTree){
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System.out.println("key:"+subTree.key+"--name:"+subTree.data);;
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traverse(subTree.leftChild);
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traverse(subTree.rightChild);
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}
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public void preOrder(TreeNode subTree){
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if(subTree!=null){
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visted(subTree);
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preOrder(subTree.leftChild);
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preOrder(subTree.rightChild);
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}
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}
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public void inOrder(TreeNode subTree){
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if(subTree!=null){
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inOrder(subTree.leftChild);
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visted(subTree);
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inOrder(subTree.rightChild);
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}
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}
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public void postOrder(TreeNode subTree) {
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if (subTree != null) {
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postOrder(subTree.leftChild);
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postOrder(subTree.rightChild);
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visted(subTree);
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}
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}
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public void nonRecPreOrder(TreeNode p){
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Stack<TreeNode> stack=new Stack<TreeNode>();
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TreeNode node=p;
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while(node!=null||stack.size()>0){
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while(node!=null){
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visted(node);
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stack.push(node);
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node=node.leftChild;
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}
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<span abp="507" style="font-size:14px;">while</span>(stack.size()>0){
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node=stack.pop();
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node=node.rightChild;
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}
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}
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}
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public void nonRecInOrder(TreeNode p){
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Stack<TreeNode> stack =new Stack<BinaryTree.TreeNode>();
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TreeNode node =p;
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while(node!=null||stack.size()>0){
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while(node!=null){
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stack.push(node);
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node=node.leftChild;
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}
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if(stack.size()>0){
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node=stack.pop();
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visted(node);
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node=node.rightChild;
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}
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}
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}
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public void noRecPostOrder(TreeNode p){
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Stack<TreeNode> stack=new Stack<BinaryTree.TreeNode>();
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TreeNode node =p;
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while(p!=null){
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for(;p.leftChild!=null;p=p.leftChild){
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stack.push(p);
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}
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while(p!=null&&(p.rightChild==null||p.rightChild==node)){
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visted(p);
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node =p;
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if(stack.empty())
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return;
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p=stack.pop();
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}
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stack.push(p);
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p=p.rightChild;
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}
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}
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public void visted(TreeNode subTree){
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subTree.isVisted=true;
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System.out.println("key:"+subTree.key+"--name:"+subTree.data);;
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}
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private class TreeNode{
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private int key=0;
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private String data=null;
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private boolean isVisted=false;
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private TreeNode leftChild=null;
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private TreeNode rightChild=null;
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public TreeNode(){}
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public TreeNode(int key,String data){
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this.key=key;
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this.data=data;
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this.leftChild=null;
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this.rightChild=null;
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}
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}
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public static void main(String[] args) {
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BinaryTree bt = new BinaryTree();
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bt.createBinTree(bt.root);
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System.out.println("the size of the tree is " + bt.size());
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System.out.println("the height of the tree is " + bt.height());
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System.out.println("*******(前序遍歷)[ABDECF]遍歷*****************");
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bt.preOrder(bt.root);
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System.out.println("*******(中序遍歷)[DBEACF]遍歷*****************");
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bt.inOrder(bt.root);
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System.out.println("*******(後序遍歷)[DEBFCA]遍歷*****************");
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bt.postOrder(bt.root);
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System.out.println("***非遞歸實現****(前序遍歷)[ABDECF]遍歷*****************");
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bt.nonRecPreOrder(bt.root);
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System.out.println("***非遞歸實現****(中序遍歷)[DBEACF]遍歷*****************");
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bt.nonRecInOrder(bt.root);
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System.out.println("***非遞歸實現****(後序遍歷)[DEBFCA]遍歷*****************");
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bt.noRecPostOrder(bt.root);
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}
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}
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</span>
輸出結果
the size of the tree is 6
the height of the tree is 3
*******(前序遍歷)[ABDECF]遍歷*****************
key:1--name:rootNode(A)
key:2--name:B
key:4--name:D
key:5--name:E
key:3--name:C
key:6--name:F
*******(中序遍歷)[DBEACF]遍歷*****************
key:4--name:D
key:2--name:B
key:5--name:E
key:1--name:rootNode(A)
key:3--name:C
key:6--name:F
*******(後序遍歷)[DEBFCA]遍歷*****************
key:4--name:D
key:5--name:E
key:2--name:B
key:6--name:F
key:3--name:C
key:1--name:rootNode(A)
***非遞歸實現****(前序遍歷)[ABDECF]遍歷*****************
key:1--name:rootNode(A)
key:2--name:B
key:4--name:D
key:5--name:E
key:3--name:C
key:6--name:F
***非遞歸實現****(中序遍歷)[DBEACF]遍歷*****************
key:4--name:D
key:2--name:B
key:5--name:E
key:1--name:rootNode(A)
key:3--name:C
key:6--name:F
***非遞歸實現****(後序遍歷)[DEBFCA]遍歷*****************
key:4--name:D
key:5--name:E
key:2--name:B
key:6--name:F
key:3--name:C
key:1--name:rootNode(A)