Chapter 10 Trees &Binary Trees

Chapter 10Trees &Binary Trees • Binary Trees • Definitions, Properties, implementation • Traversal of Binary Trees & It’s Applications • Trees & Forests vs. Binary Trees • Huffman Tree (哈夫曼树） • Binary Search Trees • Height Balance: AVL Trees

Trees • A tree is either : (rooted) • Empty • Contains a root and N subtrees (N>=0). A tree is a nonlinear data structure. root subtree

Terms • Child • Parent • Ancestor • Sibling : nodes that share a common parent. The only item which has no Parent.

Terms (continue) • Leaf : a node with no children.(end vertices or external vertices)The remaining vertices are called the internal vertices. • Node depth : the number of links on the path from the root to the node. • Tree height :the number of vertices in the longest path that occurs. (page 286) the depth of the deepest node. An empty tree has height 0. Two rules about parents: • The root has no parent. • Every other node has exactly one parent.

结点(node/vertex) • 结点的度(degree) • 分支(branch)结点 • 叶(leaf)结点 • 子女(child)结点 • 双亲(parent)结点 • 兄弟(sibling)结点 • 祖先(ancestor)结点 • 子孙(descendant)结点 • 结点所处层次(level) • 树的高度(depth) • 树的度(degree) • 有序树 • 无序树 • 森林

Binary Trees Different binary tree

2-Trees (p290) • As 2-treeis a tree in which every vertex except the leaves has exactly two children.

Full Binary Trees • A binary tree t is full if t is a two-tree with all its leaves one the same level. a binary tree t is full if t is empty or t’s left and right subtrees have the same height and both are full. Full Binary Tree

Complete Binary Trees • A binary tree t is complete if t is full through level height(t)-1, and all the leaves at lowest level occupy the leftmost positions in the tree. Complete Binary Tree

The Binary Tree Theorem(Properties) • N(i)<=2^(i-1) • If height(t)=k, N(t)<=2^k –1; • If t is a complete binary tree of n nodes, height(t) =

二叉树的性质 性质1若二叉树的层次从0开始, 则在二叉树的第 i 层最多有 2i 个结点。(i 0) [证明用数学归纳法] 性质2高度为k的二叉树最多有 2k+1-1个结点。 (k -1) [证明用求等比级数前k项和的公式] 性质3对任何一棵二叉树, 如果其叶结点个数为 n0, 度为2的非叶结点个数为 n2, 则有 n0＝n2＋1

证明：若设度为1的结点有n1个，总结点个数为n，总边数为e，则根据二叉树的定义，证明：若设度为1的结点有n1个，总结点个数为n，总边数为e，则根据二叉树的定义， n = n0 + n1 + n2e = 2n2 + n1 = n - 1 因此，有 2n2 + n1= n0 + n1 + n2 - 1 n2 = n0 - 1 n0 = n2 + 1 定义1满二叉树(Full Binary Tree) 定义2完全二叉树(Complete Binary Tree) 若设二叉树的高度为h，则共有h+1层。除第h层外，其它各层(0h-1)的结点数都达到最大个数，第h层从右向左连续缺若干结点，这就是完全二叉树。

性质4具有n个结点的完全二叉树的高度为 log2(n+1)-1 证明：设完全二叉树的高度为h，则有 2h- 1 < n 2h+1- 1 2h < n+1  2h+1 取对数 h < log2(n+1)  h+1

性质5如果将一棵有n个结点的完全二叉树自顶向下，同一层自左向右连续给结点编号0, 1, 2, …, n-1，然后按此结点编号将树中各结点顺序地存放于一个一维数组中, 并简称编号为i的结点为结点i (0 i n-1。则有以下关系： • 若i == 0, 则 i 无双亲若i > 0, 则 i 的双亲为(i -1)/2 • 若2*i+1 < n, 则 i 的左子女为2*i+1 若2*i+2 < n, 则 i 的右子女为2*i+2 • 若 i 为偶数, 且i != 0, 则其左兄弟为i-1 若 i 为奇数, 且i != n-1, 则其右兄弟为i+1 • i 所在层次为 log2 (i+1)

Contiguous Binary Trees

Binary tree

Linked Binary Trees

Linked implementation of binary tree:

Binary tree class: template <class Entry> class Binary tree { public: // Add methods here. protected: // Add auxiliary function prototypes here. Binary node<Entry> *root; };

Binary node class: template <class Entry> struct Binary node { // data members: Entry data; Binary node<Entry> *left; Binary node<Entry> *right; // constructors: Binary node( ); Binary node(const Entry &x); };

Constructor & Empty template <class Entry> Binary tree<Entry> :: Binary tree( ) /* Post: An empty binary tree has been created. */ { root = NULL; } template <class Entry> bool Binary tree<Entry> :: empty( ) const /* Post: A result of true is returned if the binary tree is empty. Otherwise,false is returned. */ { return root == NULL; }

Binary Tree Class Specification template <class Entry> class Binary tree { public: Binary tree( ); bool empty( ) const; void preorder(void (*visit)(Entry &)); void inorder(void (*visit)(Entry &)); void postorder(void (*visit)(Entry &)); int size( ) const; void clear( ); int height( ) const; void insert(const Entry &); Binary tree (const Binary tree<Entry> &original); Binary tree & operator = (const Binary tree<Entry> &original); Binary tree( ); protected: // Add auxiliary function prototypes here. Binary node<Entry> *root; };

Traversal of Binary Trees • At a given node there are three tasks to do in some order: Visit the node itself (V); traverse its left subtree (L); traverse its right subtree (R). There are six ways to arrange these tasks: • V L R L V R L R V • V R L R V L R L V.

Traversal of Binary Trees • By standard convention, these are reduced to three by considering only the ways in which the left subtree is traversed before the right.

Inorder traversal template <class Entry> void Binary tree<Entry> :: recursive_inorder(Binary node<Entry> *sub_root, void (*visit)(Entry &)) { if (sub_root != NULL) { recursive_inorder(sub_root ->left, visit); (*visit)(sub_root ->data); recursive_inorder(sub_root ->right, visit); } }

Preorder traversal template <class Entry> void Binary tree<Entry> :: recursive_preorder(Binary node<Entry> *sub_root, void (*visit)(Entry &)) { if (sub_root != NULL) { (*visit)(sub_root ->data); recursive_preorder(sub_root ->left, visit); recursive_preorder(sub_root ->right, visit); } }

Postorder traversal template <class Entry> void Binary tree<Entry> :: recursive_postorder(Binary node<Entry> *sub_root, void (*visit)(Entry &)) { if (sub_root != NULL) { recursive_postorder(sub_root ->left, visit); recursive_postorder(sub_root ->right, visit); (*visit)(sub_root ->data); } }

Expression Trees

Traversal of Comparison tree

Applications of Traversal of Binary tree Size and height template <class Entry > int BinaryTree< Entry >:: Size ( constBinTreeNode < Entry > *t ) const { if ( t == NULL ) return 0; elsereturn 1 + Size ( t→left ) + Size ( t→right ); } template <class Entry > intBinaryTree< Entry >:: Height ( const BinTreeNode < Entry > *t ) const { if ( t == NULL ) return0; else return 1 + Max (Height ( t→left ), Height ( t→right ) ); }

Other Applications of Traversal • Copy • Interchange • all left and right subtrees • Width • Traversal level by level

Exercises • P442 • 10.1 E3 • E11, E12 • Binary_tree Create()

Homework • p441 - 442 • Exercise 10.1 E2, E5

Chapter 10 Trees &Binary Trees