1. Experimental tasks
Using binary linked lists as storage structures, write algorithms for traversing binary trees in preorder, inorder, postorder and hierarchical order.
2. Experimental questions
Assume that the binary tree is stored as a binary chain, each node value is a single character and all node values are different. Write an experimental program to construct a binary chain from the in-order sequence and the post-order sequence of the binary tree, implement the search, height, pre-order traversal, in-order traversal, post-order traversal and level traversal algorithms, and test it with relevant data.
3. Experimental process:
Import the required data from collections import deque. In a binary tree, the node structure of the standard storage method is (data, lchild, rchild). This method is usually called a binary chain.
from collections import deque
class BTNode: #二叉链中结点类
def __init__(self,d=None): #构造方法
self.data=d #结点值
self.lchild=None #左孩子指针
self.rchild=None
Next is the implementation of the basic operation algorithm of binary trees and the design of binary tree classes. Here I designed the root node b as the unique identifier of the binary tree, defined the brackets of the binary chain to represent the string, and searched for the value x (using a recursive algorithm) and height (Height).
class BTree: #二叉树类
def __init__(self,d=None): #构造方法
self.b=None #根结点指针
def DispBTree(self): #返回二叉链的括号表示串
return self._DispBTree1(self.b)
def _DispBTree1(self,t): #被DispBTree方法调用
if t==None: #空树返回空串
return ""
else:
bstr=t.data #输出根结点值
if t.lchild!=None or t.rchild!=None:
bstr+="(" #有孩子结点时输出"("
bstr+=self._DispBTree1(t.lchild) #递归输出左子树
if t.rchild!=None:
bstr+="," #有右孩子结点时输出","
bstr+=self._DispBTree1(t.rchild) #递归输出右子树
bstr+=")" #输出")"
return bstr
def FindNode(self,x): #查找值为x的结点算法
return self._FindNode1(self.b,x)
def _FindNode1(self,t,x): #被FindNode方法调用
if t==None:
return None #t为空时返回null
elif t.data==x:
return t #t所指结点值为x时返回t
else:
p=self._FindNode1(t.lchild,x) #在左子树中查找
if p!=None:
return p #在左子树中找到p结点，返回p
else:
return self._FindNode1(t.rchild,x) #返回在右子树中查找结果
def Height(self): #求二叉树高度的算法
return self._Height1(self.b)
def _Height1(self,t): #被Height方法调用
if t==None:
return 0 #空树的高度为0
else:
lh=self._Height1(t.lchild) #求左子树高度lchildh
rh=self._Height1(t.rchild) #求右子树高度rchildh
return max(lh,rh)+1
Next, design the traversal of the binary tree in preorder, midorder, and postorder. Binary tree traversal is to visit all the nodes in the binary tree in a certain order, and each node is visited only once. The
traversal order is first order, middle order, and last order.
def PreOrder(bt): #先序遍历的递归算法
_PreOrder(bt.b)
def _PreOrder(t): #被PreOrder方法调用
if t!=None:
print(t.data,end=' ') #访问根结点
_PreOrder(t.lchild) #先序遍历左子树
_PreOrder(t.rchild) #先序遍历右子树
def InOrder(bt): #中序遍历的递归算法
_InOrder(bt.b)
def _InOrder(t): #被InOrder方法调用
if t!=None:
_InOrder(t.lchild) #中序遍历左子树
print(t.data,end=' ') #访问根结点
_InOrder(t.rchild) #中序遍历右子树
def PostOrder(bt): #后序遍历的递归算法
_PostOrder(bt.b)
def _PostOrder(t): #被PostOrder方法调用
if t!=None:
_PostOrder(t.lchild) #后序遍历左子树
_PostOrder(t.rchild) #后序遍历右子树
print(t.data,end=' ')
Construction of a binary tree: A binary chain is constructed from the post-order sequence posts and the mid-order sequence ins.
def LevelOrder(bt): #层次遍历的算法
qu=deque() #将双端队列作为普通队列qu
qu.append(bt.b) #根结点进队
while len(qu)>0: #队不空循环
p=qu.popleft() #出队一个结点
print(p.data,end=' ') #访问p结点
if p.lchild!=None: #有左孩子时将其进队
qu.append(p.lchild)
if p.rchild!=None: #有右孩子时将其进队
qu.append(p.rchild)
def CreateBTree2(posts,ins): #由后序序列posts和中序序列ins构造二叉链
bt=BTree()
bt.b=_CreateBTree2(posts,0,ins,0,len(posts))
return bt
def _CreateBTree2(posts,i,ins,j,n):
if n<=0: return None
d=posts[i+n-1] #取后序序列尾元素d
t=BTNode(d) #创建根结点(结点值为d)
p=ins.index(d) #在ins中找到根结点的索引
k=p-j #确定左子树中结点个数k
t.lchild=_CreateBTree2(posts,i,ins,j,k) #递归构造左子树
t.rchild=_CreateBTree2(posts,i+k,ins,p+1,n-k-1) #递归构造右子树
return t
Main program design
#主程序
ins=['D','G','B','A','E','C','F']
posts=['G','D','B','E','F','C','A']
print()
print(" 中序:",end=' '); print(ins)
print(" 后序:",end=' '); print(posts)
print(" 构造二叉树bt")
bt=BTree()
bt=CreateBTree2(posts,ins)
print(" bt:",end=' '); print(bt.DispBTree())
x='X'
p=bt.FindNode(x)
if p!=None: print(" bt中存在"+x)
else: print(" bt中不存在"+x)
print(" bt的高度=%d" %(bt.Height()))
print(" 先序序列:",end=' ');PreOrder(bt);print()
print(" 中序序列:",end=' ');InOrder(bt);print()
print(" 后序序列:",end=' ');PostOrder(bt);print()
print(" 层次序列:",end=' ');LevelOrder(bt);print()
Output result:
Overall code:
from collections import deque
class BTNode: #二叉链中结点类
def __init__(self,d=None): #构造方法
self.data=d #结点值
self.lchild=None #左孩子指针
self.rchild=None #右孩子指针
class BTree: #二叉树类
def __init__(self,d=None): #构造方法
self.b=None #根结点指针
def DispBTree(self): #返回二叉链的括号表示串
return self._DispBTree1(self.b)
def _DispBTree1(self,t): #被DispBTree方法调用
if t==None: #空树返回空串
return ""
else:
bstr=t.data #输出根结点值
if t.lchild!=None or t.rchild!=None:
bstr+="(" #有孩子结点时输出"("
bstr+=self._DispBTree1(t.lchild) #递归输出左子树
if t.rchild!=None:
bstr+="," #有右孩子结点时输出","
bstr+=self._DispBTree1(t.rchild) #递归输出右子树
bstr+=")" #输出")"
return bstr
def FindNode(self,x): #查找值为x的结点算法
return self._FindNode1(self.b,x)
def _FindNode1(self,t,x): #被FindNode方法调用
if t==None:
return None #t为空时返回null
elif t.data==x:
return t #t所指结点值为x时返回t
else:
p=self._FindNode1(t.lchild,x) #在左子树中查找
if p!=None:
return p #在左子树中找到p结点，返回p
else:
return self._FindNode1(t.rchild,x) #返回在右子树中查找结果
def Height(self): #求二叉树高度的算法
return self._Height1(self.b)
def _Height1(self,t): #被Height方法调用
if t==None:
return 0 #空树的高度为0
else:
lh=self._Height1(t.lchild) #求左子树高度lchildh
rh=self._Height1(t.rchild) #求右子树高度rchildh
return max(lh,rh)+1
def PreOrder(bt): #先序遍历的递归算法
_PreOrder(bt.b)
def _PreOrder(t): #被PreOrder方法调用
if t!=None:
print(t.data,end=' ') #访问根结点
_PreOrder(t.lchild) #先序遍历左子树
_PreOrder(t.rchild) #先序遍历右子树
def InOrder(bt): #中序遍历的递归算法
_InOrder(bt.b)
def _InOrder(t): #被InOrder方法调用
if t!=None:
_InOrder(t.lchild) #中序遍历左子树
print(t.data,end=' ') #访问根结点
_InOrder(t.rchild) #中序遍历右子树
def PostOrder(bt): #后序遍历的递归算法
_PostOrder(bt.b)
def _PostOrder(t): #被PostOrder方法调用
if t!=None:
_PostOrder(t.lchild) #后序遍历左子树
_PostOrder(t.rchild) #后序遍历右子树
print(t.data,end=' ') #访问根结点
def LevelOrder(bt): #层次遍历的算法
qu=deque() #将双端队列作为普通队列qu
qu.append(bt.b) #根结点进队
while len(qu)>0: #队不空循环
p=qu.popleft() #出队一个结点
print(p.data,end=' ') #访问p结点
if p.lchild!=None: #有左孩子时将其进队
qu.append(p.lchild)
if p.rchild!=None: #有右孩子时将其进队
qu.append(p.rchild)
def CreateBTree2(posts,ins): #由后序序列posts和中序序列ins构造二叉链
bt=BTree()
bt.b=_CreateBTree2(posts,0,ins,0,len(posts))
return bt
def _CreateBTree2(posts,i,ins,j,n):
if n<=0: return None
d=posts[i+n-1] #取后序序列尾元素d
t=BTNode(d) #创建根结点(结点值为d)
p=ins.index(d) #在ins中找到根结点的索引
k=p-j #确定左子树中结点个数k
t.lchild=_CreateBTree2(posts,i,ins,j,k) #递归构造左子树
t.rchild=_CreateBTree2(posts,i+k,ins,p+1,n-k-1) #递归构造右子树
return t
#主程序
ins=['D','G','B','A','E','C','F']
posts=['G','D','B','E','F','C','A']
print()
print(" 中序:",end=' '); print(ins)
print(" 后序:",end=' '); print(posts)
print(" 构造二叉树bt")
bt=BTree()
bt=CreateBTree2(posts,ins)
print(" bt:",end=' '); print(bt.DispBTree())
x='X'
p=bt.FindNode(x)
if p!=None: print(" bt中存在"+x)
else: print(" bt中不存在"+x)
print(" bt的高度=%d" %(bt.Height()))
print(" 先序序列:",end=' ');PreOrder(bt);print()
print(" 中序序列:",end=' ');InOrder(bt);print()
print(" 后序序列:",end=' ');PostOrder(bt);print()
print(" 层次序列:",end=' ');LevelOrder(bt);print()