章节主题

• 二叉树及其应用

• 二叉搜索树

• AVL树

• 优先级队列

Chapter Topics

• Binary Trees and Their Applications 

• Binary Search Trees

• AVLTrees

• Priority Queues

二叉树及其应用（Binary Trees and Their Applications）

二叉树

• 二叉树就像一个链表，除了每个节点最多可以有两个后继。

• 二叉树中节点X的后继者称为X的子节点。

• 在二叉树中，每个节点最多只有一个前驱。 X的前身称为X的父节点。

• 二叉树中节点的子节点是左子节点或右子节点。

Binary Trees

• A binary tree is like a linked list, except each node may have up to two successors.

• A successor of a node X in a binary tree is called a child of X.

• In a binary tree, each node has at most one predecessor. The predecessor of X is called the parent of X.

• A child of a node in a binary tree is either a left child or a right child.

关于二叉树的事实

• 如果节点C是另一个节点P的子节点，则P称为C的父节点。

• 二叉树可能为空。

• 非空二叉树具有唯一没有父节点的节点。 此节点称为二叉树的根。

• 没有子节点的节点称为叶子。

Facts About Binary Trees

• If a node C is a child of another node P, then P is called the parent of C.

• A binary tree may be empty.

• A nonempty binary tree has a unique node that has no parent. This node is called the root of the binary tree.

• A node with no children is called a leaf.

二叉树

• C是A的右子节点。

• E是C的左子节点。

• D和G是叶子。

• A是根。

A Binary Tree

• C is the right child of A.

• E is the left child of C.

• D and G are leaves.

• A is the root.

节点的后代

设X是二叉树T中的节点。如果Y在从X到T的叶子的路径上，则节点Y是X的后代。

• C的后代是E，F和G.

• B的后代是D.

Descendants of a Node

Let X be a node in a binary tree T. A node Y is a descendant of X if Y is on the path from X to a leaf of T.

• Descendants of C are E, F, and G.

• Descendants of B is D.

子树

• 节点X的所有后代的集合形成二叉树，称为以X为根的T的子树。

• 如果R是T的根，则以R的左子节为根的子树称为T的左子树，以R为右子节的子树称为T的右子树。

Subtrees

• The collection of all descendants of a node X forms a binary tree, called the subtree of T rooted at X.

• If R is the root of T, then the subtree rooted at the left child of R is called the left subtree of T, and the subtree rooted at the right child of R is called the right subtree of T.

二叉树的应用

• 二叉树用于组织信息以支持快速搜索。

• 二进制树的推广在数据库系统中用于存储数据。

• 二叉树用于表示算术表达式。

Applications of Binary Trees

• Binary trees are used to organize information to support fast search.

• Generalizations of binary trees are used in database systems to store data.

• Binary trees are used to represent arithmetic expressions.

ADT二叉树的实现

• 实现基于类似于链接列表中使用的Node类。

• SeeTreeNode.java。

Implementation of ADT Binary Trees

• Implementation is based on a Node class similar to what is used in linked lists.

• SeeTreeNode.java.

表示二叉树

• 二叉树由对其根节点的引用表示。

节点根;

• 空二进制树用引用表示，其值为null。

Representing a Binary Tree

• A binary tree is represented by a reference to its root node.

Node root;

• An empty binary tree is represented with a reference whose value is null.

ADT二叉树：

ADT二叉树的基本操作

• 作为抽象数据类型，二叉树具有添加和删除节点和子树的操作。 通过使用这些基本操作，您可以构建任何二叉树。

• 其他操作在树的根目录中设置或检索数据，并确定树是否为空。

• 遍历操作访问二叉树中的每个节点，使用它或对其执行某些操作。 三个标准命令是前序，中序和后序。

The ADT Binary Tree:

Basic Operations of the ADT Binary Tree

• As an abstract datatype, the binary tree has operations that add and remove nodes and subtrees. By using these basic operations, you can build any binary tree.

• Other operations set or retrieve the data in the root of the tree and determine whether the tree is empty.

• Traversal operations visit every node in a binary tree, do something with it or to it. Three standard orders are preorder, inorder, and postorder.

ADT二叉树：

ADT二叉树的基本操作

• 可用于特定ADT二叉树的操作取决于要实现的二叉树的类型

• ADT二叉树的基本操作

The ADT Binary Tree:

Basic Operations of the ADT Binary Tree

• The operations available for a particular ADT binary tree depend on the type of binary tree being implemented

• Basic operations of the ADT binary tree

• • createBinaryTree()

  • createBinaryTree(rootItem)

  • makeEmpty()

  • isEmpty()

  • getRoot() throws TreeException 

  • setRoot(rootItem) throws

            UnsupportedOperationException

• ADT二叉树的常规操作（附加操作）

• General operations (additional operations) of the ADT binary tree

  • createBinaryTree (rootItem, leftTree, rightTree) 

  • setRoot(newItem)

  • attachLeft(newItem) throws TreeException

  • attachRight(newItem) throws TreeException

  • attachLeftSubtree(leftTree) throws TreeException

  • attachRightSubtree(rightTree) throws TreeException 

  • detachLeftSubtree() throws TreeException

  • detachRightSubtree() throws TreeException

• 二叉树的遍历

• 二叉树的遍历算法访问树中的每个节点

• 递归遍历算法 

  • 前序遍历

  • 中序遍历

  • 后序遍历

• 这些遍历中的每一个都只访问二叉树中的每个节点一次，执行相同的操作（O（1）），独立于n，遍历为O（n）

• Traversals of a Binary Tree

• A traversal algorithm for a binary tree visits each node in the tree

• Recursive traversal algorithms 

  • Preorder traversal

  • Inorder traversal

  • Postorder traversal

• Each of these traversals visits every node in a binary tree exactly once, performs the same operation (O(1)),independently of n, and traversal is O(n)

• 前序遍历算法假设“访问节点”表示显示节点的数据项。

if（树不为空）{

    显示树根中的数据

    以前序遍历（树根的左子树）

    以前序遍历（（树的根的右子树）

}

• Preorder traversal algorithm Assumes that “visit a node” means to display the node’s data item.

if (the tree is not empty) {

    Display the data in the root of the tree

    preorder(Left subtree of the tree ‘s root) 

    preorder(Right subtree of the tree ‘s root)

}

• 中序遍历算法

假设“访问节点”意味着显示节点的数据项。

if（树不为空）{

    中序（树根的左子树）

    显示树根中的数据

    中序（树根的右子树）

}

• Inorder traversal algorithm 

Assumes that “visit a node” means to display the node’s data item.

if (the tree is not empty) {

    Inorder(Left subtree of the tree ‘s root) 

    Display the data in the root of the tree

    Inorder(Right subtree of the tree ‘s root)

 }

• 后序遍历算法

假设“访问节点”意味着显示节点的数据项。

if（树不为空）{

后序（树根的左子树）

后序（树根的右子树）

显示树根中的数据

}

• Post-order traversal algorithm

Assumes that “visit a node” means to display the node’s data item.

if (the tree is not empty) {

Postorder(Left subtree of the tree ‘s root)

Postorder(Right subtree of the tree ‘s root)

Display the data in the root of the tree

}

• 二叉树的可能表示

  • 基于数组的表示

  • Java类用于在树中定义节点

  • 使用树节点数组表示二叉树

  • 每个树节点包含一个数据部分和两个索引（每个节点的子节点一个）

  • 需要创建一个跟踪可用节点的空闲列表

• Possible Representations of a Binary Tree

  • An array-based representation

  • A Java class is used to define a node in the tree

  • A binary tree is represented by using an array of tree nodes

  • Each tree node contains a data portion and two indexes (one for each of the node’s children)

  • Requires the creation of a free list which keeps track of available nodes

• 基于数组的完整树表示

  • 如果二叉树已完成并仍保持完整

    • 可以使用基于阵列的内存高效实现

• An array-based representation of a complete tree

  • If the binary tree is complete and remains complete

    • A memory-efficient array-based implementation can be used

• 基于参考的表示

  • Java引用可用于链接树中的节点

• A reference-based representation

  • Java references can be used to link the nodes in the tree

基于参考的ADT二叉树实现

• 为ADT二叉树提供基于引用的实现的类

  • TreeNode

    • 表示二叉树中的节点

  • TreeException

    • 异常类

  • BinaryTreeBasis

    • 基本树操作的抽象类

  • BinaryTree

    • 提供二叉树的常规操作

    • 扩展BinaryTreeBasis

A Reference-Based Implementation of the ADT Binary Tree

• Classes that provide a reference-based implementation for the ADT binary tree

  • TreeNode

    • Represents a node in a binary tree

  • TreeException

    • An exception class

  • BinaryTreeBasis

    • An abstract class of basic tree operation

  • BinaryTree

    • Provides the general operations of a binary tree 

    • Extends BinaryTreeBasis

使用迭代器的树遍历

• TreeIterator

  • 实现Java Iterator接口

    • 必须提供三种方法：next，hasNext和remove

    • 未实现删除

      • 删除节点取决于树的类型

      • BinaryTreeBasis没有删除节点的方法。

• 提供将迭代器设置为所需遍历类型的方法

• 使用队列来维护树中节点的当前遍历

Tree Traversals Using an Iterator

• TreeIterator

  • Implements the Java Iterator interface

  • must provide three methods: next, hasNext, and remove

  • Not implement remove

    • Removal of a node depends on the type of tree

    • BinaryTreeBasis doesn’t have a method removing a node.

• Provides methods to set the iterator to the type of traversal desired

• Uses a queue to maintain the current traversal of the nodes in the tree

使用迭代器的树遍历

• 非递归遍历（可选）

  • 迭代方法和显式堆栈可用于模拟从递归调用返回到中序的操作

Tree Traversals Using an Iterator

• Nonrecursive traversal (optional)

• •  An iterative method and an explicit stack can be used to mimic actions at a return from a recursive call to inorder