A. Binary tree traversal

1. Preorder traversal of the binary tree:

(1) If the binary tree is empty, it is a no-op operation and returns empty.
(2) Visit the root node.
(3) Preorder traversal of the left subtree.
(4) Preorder traverse the right subtree.

##a. Recursive algorithm for pre-order traversal of binary trees:

void PreOrderTraverse(BiTree BT) { if(BT) { printf("%c",BT->data); //访问根结点 PreOrderTraverse(BT->lchild); //前序遍历左子树 PreOrderTraverse(BT->rchild); //前序遍历右子树 } }

b. Non-recursive algorithm for binary tree pre-order traversal using stack to store the right subtree of each node:

( 1) When the tree is empty, point the pointer p to the root node, and p is the current node pointer.
(2) First visit the current node p and push p into the stack S.
(3) Let p point to its left child.
(4) Repeat steps (2) and (3) until p is empty.
(5) Pop the top element from stack S and point p to the right child of this element.
(6) Repeat steps (2)~(5) until p is empty and stack S is also empty.
(7) The traversal ends.
Non-recursive algorithm using pre-order traversal of the stack:

void PreOrderNoRec(BiTree BT) { stack S; BiTree p=BT->root; while((NULL!=p)||!StackEmpty(S)) { if(NULL!=p) { printf("%c",p->data); Push(S,p); p=p->lchild; } else { p=Top(S); Pop(S); p=p->rchild; } } }

c. Use binary linked list storage Non-recursive algorithm for pre-order traversal of a binary tree:

void PreOrder(pBinTreeNode pbnode) { pBinTreeNode stack[100]; pBinTreeNode p; int top; top=0; p=pbnode; do { while(p!=NULL) { printf("%d\n",p->data); //访问结点p top=top+1; stack[top]=p; p=p->llink; //继续搜索结点p的左子树 } if(top!=0) { p=stack[top]; top=top-1; p=p->rlink; //继续搜索结点p的右子树 } }while((top!=0)||(p!=NULL)); }

void InOrderTraverse(BiTree BT) { if(BT) { InOrderTraverse(BT->lchild); //中序遍历左子树 printf("%c",BT->data); //访问根结点 InOrderTraverse(BT->rchild); //中序遍历右子树 } }

b. Non-recursive algorithm for in-order traversal of binary trees using stack storage:

(1) When the tree When it is empty, point the pointer p to the root node, and p is the current node pointer.

(2) Push p onto stack S and make p point to its left child.
(3) Repeat step (2) until p is empty.
(4) Pop the top element from stack S and point p to this element.
(5) Access the current node p and point p to its right child.
(6) Repeat steps (2)~(5) until p is empty and stack S is also empty.
(7) The traversal ends.
Non-recursive algorithm using in-order traversal of the stack:

void IneOrderNoRec(BiTree BT) { stack S; BiTree p=BT->root; while((NULL!=p)||!StackEmpty(S)) { if(NULL!=p) { Push(S,p); p=p->lchild; } else { p=Top(S); Pop(S); printf("%c",p->data); p=p->rchild; } } }

c. Use binary fork Non-recursive algorithm for in-order traversal of a binary tree stored in a linked list:

void InOrder(pBinTreeNode pbnode) { pBinTreeNode stack[100]; pBinTreeNode p; int top; top=0; p=pbnode; do { while(p!=NULL) { top=top+1; stack[top]=p; //结点p进栈 p=p->llink; //继续搜索结点p的左子树 } if(top!=0) { p=stack[top]; //结点p出栈 top=top-1; printf("%d\n",p->data); //访问结点p p=p->rlink; //继续搜索结点p的右子树 } }while((top!=0)||(p!=NULL)); }

3. Post-order traversal of a binary tree:

(1) If the binary tree is empty, it is a no-op operation and returns empty.

(2) Post-order traversal of the left subtree.
(3) Post-order traversal of the right subtree.
(4) Visit the root node.

a.二叉树后序遍历的递归算法：

void PostOrderTraverse(BiTree BT) { if(BT) { PostOrderTraverse(BT->lchild); //后序遍历左子树 PostOrderTraverse(BT->rchild); //后序遍历右子树 printf("%c",BT->data); //访问根结点 } }

b.使用栈存储的二叉树后序遍历的非递归算法：

算法思想：首先扫描根结点的所有左结点并入栈，然后出栈一个结点，扫描该结点的右结点并入栈，再扫描该右结点的所有左结点并入栈，当一个结点的左、右子树均被访问后再访问该结点。因为在递归算法中，左子树和右子树都进行了返回，因此为了区分这两种情况，还需要设置一个标识栈tag，当tag的栈顶元素为0时表示从左子树返回，为1表示从右子树返回。
(1)当树为空时，将指针p指向根结点，p为当前结点指针。
(2)将p压入栈S中，0压入栈tag中，并令p指向其左孩子。
(3)重复执行步骤(2)，直到p为空。
(4)如果tag栈中的栈顶元素为1，跳至步骤(6)。
(5)如果tag栈中的栈顶元素为0，跳至步骤(7)。
(6)将栈S的栈顶元素弹出，并访问此结点，跳至步骤(8)。
(7)将p指向栈S的栈顶元素的右孩子。
(8)重复执行步骤(2)~(7)，直到p为空并且栈S也为空。
(9)遍历结束。
使用栈的后序遍历非递归算法：

void PostOrderNoRec(BiTree BT) { stack S; stack tag; BiTree p=BT->root; while((NULL!=p)||!StackEmpty(S)) { while(NULL!=p) { Push(S,p); Push(tag,0); p=p->lchild; } if(!StackEmpty(S)) { if(Pop(tag)==1) { p=Top(S); Pop(S); printf("%c",p->data); Pop(tag); //栈tag要与栈S同步 } else { p=Top(S); if(!StackEmpty(S)) { p=p->rchild; Pop(tag); Push(tag,1); } } } } }

c.使用二叉链表存储的二叉树后序遍历非递归算法：

void PosOrder(pBinTreeNode pbnode) { pBinTreeNode stack[100]; //结点的指针栈 int count[100]; //记录结点进栈次数的数组 pBinTreeNode p; int top; top=0; p=pbnode; do { while(p!=NULL) { top=top+1; stack[top]=p; //结点p首次进栈 count[top]=0; p=p->llink; //继续搜索结点p的左子树 } p=stack[top]; //结点p出栈 top=top-1; if(count[top+1]==0) { top=top+1; stack[top]=p; //结点p首次进栈 count[top]=1; p=p->rlink; //继续搜索结点p的右子树 } else { printf("%d\n",p->data); //访问结点p p=NULL; } }while((top>0)); }

B 线索化二叉树：

线索化二叉树的结点结构图：

线索化二叉树的结点类型说明：

typedef struct node { DataType data; struct node *lchild, *rchild; //左、右孩子指针 int ltag, rtag; //左、右线索 }TBinTNode; //结点类型 typedef TBinTNode *TBinTree;

在线索化二叉树中，一个结点是叶子结点的充分必要条件是其左、右标志均为1.

中序线索化二叉树及其对应的线索链表如下图：

(1)中序线索化二叉树的算法：

void InOrderThreading(TBinTree p) { if(p) { InOrderThreading(p->lchild); //左子树线索化 if(p->lchild) p->ltag=0; else p->ltag=1; if(p->rchild) p->rtag=0; else p->rtag=1; if(*(pre)) //若*p的前驱*pre存在 { if(pre->rtag==1) pre->rchild=p; if(p->ltag==1) p->lchild=pre; } pre=p; //另pre是下一访问结点的中序前驱 InOrderThreading(p->rchild); //右子树线索化 } }

(2)在中序线索化二叉树下，结点p的后继结点有以下两种情况：

①结点p的右子树为空，那么p的右孩子指针域为右线索，直接指向结点p的后继结点。
②结点p的右子树不为空，那么根据中序遍历算法，p的后继必是其右子树中第1个遍历到的结点。

中序线索化二叉树求后继结点的算法：

TBinTNode *InOrderSuc(BiThrTree p) { TBinTNode *q; if(p->rtag==1) //第①情况 return p->rchild; else //第②情况 { q=p->rchild; while(q->ltag==0) q=q->lchild; return q; } }

中序线索化二叉树求前驱结点的算法：

TBinTNode *InOrderPre(BiThrTree p) { TBinTNode *q; if(p->ltag==1) return p->lchild; else { q=p->lchild; //从*p的左孩子开始查找 while(q->rtag==0) q=q->rchild; return q; } }

(3)遍历中序线索化二叉树的算法

void TraversInOrderThrTree(BiThrTree p) { if(p) { while(p->ltag==0) p=p->lchild; while(p) { printf("%c",p->data); p=InOrderSuc(p); } } }

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