Binary search Tree

two fork Find tree (binary search tree) is also known as a binary trees, an ordered binary tree (ordered binary tree), a sort of binary (sorted binary trees), refers to an empty tree or a two-fork tree with the following properties:

• If the left subtree of any node is not empty, the value of all nodes on the left subtree is less than the value of its root node;
• If the right subtree of any node is not empty, the value of all nodes on the right subtree is greater than the value of its root node;
• The left and right subtrees of any node are also two-fork lookup trees, respectively.
• There are no nodes with key values equal (no duplicate nodes).  

Supported operations:

• Traverse
• Inquire
  • Find
  • Maximum keyword element and minimum keyword element
  • Successors and precursors
• Inserting and deleting

Pseudo code:

x = T.root

Traverse

// Middle Sequence Traversal inorder-tree-Walk (x)    if  x≠nil        inorder-tree-Walk (x.left)        print X.key        inorder-tree-Walk (x.right)

Find

//Recursive versiontree-Search (x,k)ifx = = NIL or k = =X.keyreturnxifK <X.keyreturntree-Search (x.left,k)Else        returntree-Search (x.right,k)//Iteration Versioniterative-tree-Search (x,k) whileX≠nil and K≠x.keyifK <X.key x=X.leftElsex=X.rightreturnX

Maximum keyword element and minimum keyword element

// Minimum keyword element tree-Min (x)    while  x.left≠nil        = x.left    return  x  // max keyword element tree-max (x)    while  x.left≠nil        = x.right     return X

Successors and precursors

//Middle Sequence Traversal successortree-successor (x)ifX.right≠nil//X has right child, then successor is the smallest key element of its right child        returntree-Min (x.right)//x No right child, if X is the left child, then the successor is the parent node, if X is the right child, then the successor is X's left child's lowest ancestor, may not exist (NIL)y =X.P whileY≠nil and x = =y.right x=y y=Y.Preturny//Middle sequence traversal precursortree-predeccessor (x)ifX.left≠nil//X has left child, then successor is the largest key element of its left child        returntree-Max (x.left)//x No left child, if the current node is his father's right child, then the father is his predecessor y =X.P whileY≠nil and x==Y.left x=y y=Y.PreturnY

Insert

//find the insertion position, and then determine if the insertion node is left child, right child, or root.tree-Insert (t,z) y= NIL//Record z parent nodex = T.root//record the insertion position of Z     whileX≠nil y=xifZ.key <X.key x=X.leftElsex=x.right Z.P=yify = =NIL T.root=ZElse ifZ.key <Y.key Y.left=ZElseY.right= Z

Delete

//replacing subtree U with subtree V and establishing a parent-child relationship between V and U.PTransplant (T,U,V)ifU.P = =NIL T.root=vElse ifU = =U.p.left U.p.left=vElseU.p.right=vifV≠nil V.P=U.P//Delete, in three different casestree-Delete (t,z)ifZ.left = NIL//left dial hand tree is emptyTransplant (T,z,z.right)Else ifZ.right = NIL//Right sub-tree is emptyTransplant (T,z,z.left)Else//It 's two different things .y = tree-min (z.right)//successor of Z        ifY.p≠z//y not Z's right child converts it to the right childTransplant (t,y,y,right) Y.right=z.right Y.RIGHT.P=y Transplant (t,z,y) Y.left= Z.left//establishing a parent-child relationship between Y and Z.leftY.LEFT.P = y

time complexity: two fork find tree search,minimum,maximum,predecessor,successor,insert,delete time complexity t (n) =o (h), O (LG N) =

the sequential traversal of binary search tree can get the ordered sequence of a keyword.  

Binary search Tree