【数据结构】二叉树、AVL树

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二叉树

二叉树是每个结点最多有两个子树的有序树。通常子树的根被称作“左子树”(left subtree)和“右子树”(right subtree)。二叉树的每个结点至多只有二棵子树(不存在度大于2的结点),二叉树的子树有左右之分,次序不能颠倒。

二叉树有几点重要的性质:

  • 性质1:在二叉树的第 i 层上至多有2i-1 个结点。 (i≥1)
  • 性质2:深度为 k 的二叉树上至多含2k-1 个结点(k≥1)。
  • 性质3:对任何一棵二叉树,若它含有n0 个叶子结点、n2 个度为 2 的结点,则必存在关系式:n0 = n2+1。
  • 性质4:具有 n 个结点的完全二叉树的深度为log2n+1
  • 性质5:若对含 n 个结点的完全二叉树从上到下且从左至右进行 1 至 n 的编号,则对完全二叉树中任意一个编号为 i 的结点:
    (1) 若 i=1,则该结点是二叉树的根,无双亲,否则,编号为i/2 的结点为其双亲结点;
    (2) 若 2i>n,则该结点无左孩子,否则,编号为 2i 的结点为其左孩子结点;
    (3) 若 2i+1>n,则该结点无右孩子结点,否则,编号为2i+1 的结点为其右孩子结点。

采用链式存储结构实现二叉树

链式存储二叉树

1.首先我们要构造可以表示二叉树的节点的结构Binary_node

2.构造类二叉树 Binary_tree,并编写其几个基本的成员函数

Empty()-检查树是否为空;clear()-将树清空;size()-得到树的大小;leaf_count()-得到叶子数目;height()-得到树高;

以及几个重要的成员函数

  1. Binary_tree(constBinary_tree<Entry>&original);拷贝构造成员函数
  2. Binary_tree&operator=(constBinary_tree<Entry>&original);重载赋值操作符
  3. ~Binary_tree();析构函数

3.分别编写遍历算法的成员函数

    voidinorder(void(*visit)(Entry&));中序遍历(LVR)
  1. voidpreorder(void(*visit)(Entry&));前序遍历(VLR)
  2. voidpostorder(void(*visit)(Entry&));后续遍历(LRV)

因为二叉树的性质,三种遍历算法我们都用递归实现,所以分别编写其递归函数

    voidrecursive_inorder(Binary_node<Entry>*sub_root,void(*visit)(Entry&));
  1. voidrecursive_preorder(Binary_node<Entry>*sub_root,153); background-color:inherit; font-weight:bold">void(*visit)(Entry&));
  2. voidrecursive_postorder(Binary_node<Entry>*sub_root,153); background-color:inherit; font-weight:bold">void(*visit)(Entry&));

4.作为辅助,我们再编写一个print_tree的函数,用以以括号表示法输出
同样使用递归,编写递归函数void recursive_print(Binary_node<Entry>*sub_root);
几个重要的函数代码如下:

    template<classEntry>
  1. voidBinary_tree<Entry>::inorder(void(*visit)(Entry&))
  2. //Post:Thetreehasbeentraversedininordersequence
  3. //Uses:Thefunctionrecursive_inorder
  4. {
  5. recursive_inorder(root,visit);
  6. }
  7. voidBinary_tree<Entry>::recursive_inorder(Binary_node<Entry>*sub_root,0); background-color:inherit">//Pre:sub_rootiseitherNULLorpointstoasubtreeoftheBinary_tree
  8. //Post:Thesubtreehasbeentraversedininordersequence
  9. //Uses:Thefunctionrecursive_inorderrecursively
  10. {
  11. if(sub_root!=NULL){
  12. recursive_inorder(sub_root->left,108); list-style:decimal-leading-zero outside; color:inherit; line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> (*visit)(sub_root->data);
  13. recursive_inorder(sub_root->right,248); line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> }
  14. classEntry>
  15. voidBinary_tree<Entry>::preorder(void(*visit)(Entry&))
  16. //Post:Thetreehasbeentraversedinpreordersequence
  17. //Uses:Thefunctionrecursive_preorder
  18. recursive_preorder(root,visit);
  19. voidBinary_tree<Entry>::recursive_preorder(Binary_node<Entry>*sub_root,0); background-color:inherit">//Pre:sub_rootiseitherNULLorpointstoasubtreeoftheBinary_tree
  20. //Post:Thesubtreehasbeentraversedinpreordersequence
  21. //Uses:Thefunctionrecursive_preorderrecursively
  22. if(sub_root!=NULL){
  23. recursive_preorder(sub_root->left,108); list-style:decimal-leading-zero outside; color:inherit; line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> recursive_preorder(sub_root->right,153); background-color:inherit; font-weight:bold">voidBinary_tree<Entry>::postorder(//Post:Thetreehasbeentraversedinpostordersequence
  24. //Uses:Thefunctionrecursive_postorder
  25. recursive_postorder(root,153); background-color:inherit; font-weight:bold">voidBinary_tree<Entry>::recursive_postorder(Binary_node<Entry>*sub_root,0); background-color:inherit">//Pre:sub_rootiseitherNULLorpointstoasubtreefotheBinary_tree
  26. //Post:Thesubtreehasbeentraversedinpostordersequence
  27. //Uses:Thefunctionrecursive_postorderrecursively
  28. recursive_postorder(sub_root->left,108); list-style:decimal-leading-zero outside; color:inherit; line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> recursive_postorder(sub_root->right,248); line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> (*visit)(sub_root->data);
  29. voidBinary_tree<Entry>::print_tree()
  30. recursive_print(root);
  31. cout<<endl;
  32. voidBinary_tree<Entry>::recursive_print(Binary_node<Entry>*sub_root)
  33. cout<<sub_root->data;
  34. cout<<"(";
  35. recursive_print(sub_root->left);
  36. cout<<",";
  37. recursive_print(sub_root->right);
  38. cout<<")";
  39. //其他函数见源码


程序结果

插入二叉树并实现中序、前序和后序遍历

AVL树

AVL树得名于其发明者G.M.Adelson-Velsky和E.M.Landis。AVL树是一个各结点具有平衡高度的扩展的二叉搜索树。在AVL树中,任一结点的两个子树的高度差最多为1,AVL树的高度不会超过1,AVL树既有二叉搜索树的搜索效率又可以避免二叉搜索树的最坏情况(退化树)出现。

AVL树的表示与二叉搜索树类似,其操作基本相同,但插入和删除方法除外,因为它们必须不断监控结点的左右子树的相对高度,这也正是AVL树的优势所在。

实现AVL树的相关运算

1、首先我们修改结构Binary_node,增加Balance_factor用以表示节点平衡情况

2、从二叉搜索树中派生出AVL树,编写其关键的插入和删除成员函数

    Error_codeinsert(constRecord&new_data);
  1. Error_coderemove(constRecord&old_data);

3、入和删除函数我们都用递归实现
编写递归函数

    Error_codeavl_insert(Binary_node<Record>*&sub_root,
  1. constRecord&new_data,bool&taller);
  2. Error_codeavl_remove(Binary_node<Record>*&sub_root,
  3. constRecord&target,87); background-color:inherit; font-weight:bold">bool&shorter);

以及几个重要的调用函数
左旋右旋函数

    voidrotate_left(Binary_node<Record>*&sub_root);
  1. voidrotate_right(Binary_node<Record>*&sub_root);

两次旋转的左右平衡函数

    voidright_balance(Binary_node<Record>*&sub_root);
  1. voidleft_balance(Binary_node<Record>*&sub_root);

删除函数还要分别编写删除左树和删除右树的递归函数

    Error_codeavl_remove_right(Binary_node<Record>&sub_root,87); background-color:inherit; font-weight:bold">bool&shorter);
  1. Error_codeavl_remove_left(Binary_node<Record>*&sub_root,51); font-family:Arial; font-size:14.399999618530273px; line-height:26px"> 4、个重要的成员函数代码如下:

      classRecord>
    1. Error_codeAVL_tree<Record>::insert(constRecord&new_data)
    2. //Post:Ifthekeyofnew_dataisalreadyintheAVL_tree,acodeofduplicate_error
    3. //isreturned.Otherwise,acodeofsuccessisreturnedandtheRecord
    4. //new_dataisinsertedintothetreeinsuchawaythatthepropertiesofan
    5. //AVLtreearepreserved.
    6. booltaller;
    7. returnavl_insert(root,new_data,taller);
    8. classRecord>
    9. Error_codeAVL_tree<Record>::avl_insert(Binary_node<Record>*&sub_root,bool&taller)
    10. //Pre:sub_rootiseitherNULLorpointstoasubtreeoftheAVLtree
    11. //Post:Ifthekeyofnew_dataisalreadyinthesubtree,0); background-color:inherit">//new_dataisinsertedintothesubtreeinsuchawaythatthepropertiesof
    12. //anAVLtreehavebeenpreserved.Ifthesubtreeisincreaseinheight,the
    13. //parametertallerissettotrue;otherwiseitissettofalse
    14. //Uses:MethodsofstructAVL_node;functionsavl_insertrecursively,left_balance,andright_balance
    15. Error_coderesult=success;
    16. if(sub_root==NULL){
    17. sub_root=newBinary_node<Record>(new_data);
    18. taller=true;
    19. elseif(new_data==sub_root->data){
    20. result=duplicate_error;
    21. false;
    22. if(new_data<sub_root->data){//Insertinleftsubtree
    23. result=avl_insert(sub_root->left,taller);
    24. if(taller==true)
    25. switch(sub_root->get_balance()){//Changebalancefactors
    26. caseleft_higher:
    27. left_balance(sub_root);
    28. break;
    29. caseequal_height:
    30. sub_root->set_balance(left_higher);
    31. break;
    32. caseright_higher:
    33. sub_root->set_balance(equal_height);
    34. taller=false;
    35. else{//Insertinrightsubtree
    36. result=avl_insert(sub_root->right,taller);
    37. true)
    38. switch(sub_root->get_balance()){
    39. caseleft_higher:
    40. caseequal_height:
    41. sub_root->set_balance(right_higher);
    42. caseright_higher:
    43. right_balance(sub_root);
    44. false;//Rebalancingalwaysshortensthetree
    45. returnresult;
    46. voidAVL_tree<Record>::right_balance(Binary_node<Record>*&sub_root)
    47. //Pre:sub_rootpointstoasubtreeofanAVL_treethatisdoublyunbalanced
    48. //ontheright
    49. //Post:TheAVLpropertieshavebeenrestoredtothesubtree
    50. Binary_node<Record>*&right_tree=sub_root->right;
    51. switch(right_tree->get_balance()){
    52. right_tree->set_balance(equal_height);
    53. rotate_left(sub_root);
    54. cout<<"WARNING:programerrordetectedinright_balance"<<endl;
    55. Binary_node<Record>*sub_tree=right_tree->left;
    56. switch(sub_tree->get_balance()){
    57. right_tree->set_balance(right_higher);
    58. right_tree->set_balance(equal_height);
    59. sub_tree->set_balance(equal_height);
    60. rotate_right(right_tree);
    61. rotate_left(sub_root);
    62. voidAVL_tree<Record>::left_balance(Binary_node<Record>*&sub_root)
    63. Binary_node<Record>*&left_tree=sub_root->left;
    64. switch(left_tree->get_balance()){
    65. left_tree->set_balance(equal_height);
    66. rotate_right(sub_root);
    67. cout<<"WARNING:programerrordetectedinleft_balance"<<endl;
    68. Binary_node<Record>*sub_tree=left_tree->right;
    69. left_tree->set_balance(left_higher);
    70. sub_tree->set_balance(equal_height);
    71. rotate_left(left_tree);
    72. voidAVL_tree<Record>::rotate_left(Binary_node<Record>*&sub_root)
    73. //Pre:sub_rootpointstoasubtreeoftheAVL_tree.Thissubtreehas
    74. //anonemptyrightsubtree.
    75. //Post:sub_rootisresettopointtoitsformerrightchild,andthe
    76. //formersub_rootnodeistheleftchildofthenewsub_rootnode
    77. if(sub_root==NULL||sub_root->right==NULL)//impossiblecases
    78. cout<<"WARNING:programerrordetectedinrotate_left"<<endl;
    79. else{
    80. Binary_node<Record>*right_tree=sub_root->right;
    81. sub_root->right=right_tree->left;
    82. right_tree->left=sub_root;
    83. sub_root=right_tree;
    84. voidAVL_tree<Record>::rotate_right(Binary_node<Record>*&sub_root)
    85. if(sub_root==NULL||sub_root->left==NULL)
    86. cout<<"WARNING:programerrorindetectedinrotate_right"<<endl;
    87. else{
    88. Binary_node<Record>*left_tree=sub_root->left;
    89. sub_root->left=left_tree->right;
    90. left_tree->right=sub_root;
    91. sub_root=left_tree;
    92. Error_codeAVL_tree<Record>::remove(constRecord&old_data)
    93. boolshorter;
    94. returnavl_remove(root,old_data,shorter);
    95. Error_codeAVL_tree<Record>::avl_remove(Binary_node<Record>*&sub_root,
    96. bool&shorter)
    97. Binary_node<Record>*temp;
    98. if(sub_root==NULL)returnfail;
    99. if(target<sub_root->data)
    100. returnavl_remove_left(sub_root,target,153); background-color:inherit; font-weight:bold">if(target>sub_root->data)
    101. returnavl_remove_right(sub_root,153); background-color:inherit; font-weight:bold">if(sub_root->left==NULL){//Foundtarget:deletecurrentnode
    102. temp=sub_root;//Moverightsubtreeuptodeletenode
    103. sub_root=sub_root->right;
    104. deletetemp;
    105. shorter=if(sub_root->right==NULL){
    106. //Moveleftsubtreeuptodeletenode
    107. sub_root=sub_root->left;
    108. if(sub_root->get_balance()==left_higher){
    109. //Neithersubtreeisempty;deletefromthetaller
    110. temp=sub_root->left;//Findpredecessoroftargetanddeleteiffromlefttree
    111. while(temp->right!=NULL)temp=temp->right;
    112. sub_root->data=temp->data;
    113. avl_remove_left(sub_root,temp->data,shorter);
    114. temp=sub_root->right;
    115. while(temp->left!=NULL)temp=temp->left;
    116. avl_remove_right(sub_root,153); background-color:inherit; font-weight:bold">returnsuccess;
    117. Error_codeAVL_tree<Record>::avl_remove_right(Binary_node<Record>
    118. *&sub_root,108); list-style:decimal-leading-zero outside; color:inherit; line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> Error_coderesult=avl_remove(sub_root->right,shorter);
    119. if(shorter==true)switch(sub_root->get_balance()){
    120. sub_root->set_balance(equal_height);
    121. Binary_node<Record>*temp=sub_root->left;
    122. switch(temp->get_balance()){
    123. temp->set_balance(right_higher);
    124. shorter= temp->set_balance(equal_height);
    125. Binary_node<Record>*temp_right=temp->right;
    126. switch(temp_right->get_balance()){
    127. temp->set_balance(left_higher);
    128. temp_right->set_balance(equal_height);
    129. rotate_left(sub_root->left);
    130. Error_codeAVL_tree<Record>::avl_remove_left(Binary_node<Record>
    131. *&sub_root,87); background-color:inherit; font-weight:bold">bool&shorter)
    132. Error_coderesult=avl_remove(sub_root->left,248); line-height:18px; margin:0px!important; padding:0px 3px 0px 10px!important"> sub_root->set_balance(right_higher);
    133. Binary_node<Record>*temp=sub_root->right;
    134. Binary_node<Record>*temp_left=temp->left;
    135. switch(temp_left->get_balance()){
    136. sub_root->set_balance(left_higher);
    137. temp_left->set_balance(equal_height);
    138. rotate_right(sub_root->right);
    139. }


    实验结果

    实现如下功能:1)由{4,9,1,8,6,3,5,2,7}建AVL树B,并以括号表示法输出;2)删除B中关键字为8和2的结点,输出结果。

    分析总结

    采用链式存储结构实现二叉树

    1. 二叉树在插入的时候通过判断待插入数据与根节点数据的大小,若小于根数据,插入左树,反之插入右树(我们规定不许有重复元素);二叉树的存储结构常用于搜索等。但搜索的效率常依赖于树的结构。而树的结构与元素插入顺序有很大关系,用上述方法插入时若插入的元素是有序的,则得到的树与队列几乎没有区别,也起不到优化搜索的目的。
    2. 二叉树的遍历算法最为重要的一点是递归,递归使我们不必关心于具体的遍历每个节点的顺序,而只是将其看做三个部分,即左子树,根节点,右子树,具体子树的遍历仍是调用其递归函数
      3.在打印树的时候,我写的并不完美,因为对于叶子节点,理想应该不再打印括号,但我通过判断根节点不为空而调用递归函数,即只要节点有元素就会输出(,),还是表达出了树的意思,但并没有想到怎样达到简洁的效果

    实现AVL树的相关运算

    1. AVL树每插入,删除一个节点就会判断树的高度并改变相应节点的平衡因素,每当有节点不再满足AVL树的性质,立即通过旋转得到AVL树
    2. 旋转的函数及其巧妙。而插入和删除元素的函数要考虑的情况非常多,一种情况没有考虑到就可能达不到我们想要的效果函数的编写需要极大的耐心和对编程语句的熟练掌握。很多地方我是参考书上的,别人的代码我可以理解,但我自己写可能还是会漏掉很多情况。

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