Tree and two-fork tree

Tree : A finite set of n (n≥0) nodes.

In any non-empty tree: 1. When there is only one node 2.n>1 that is called the root, the remaining nodes can be divided into m (m>0) disjoint finite sets T1,t2,..., Tn,ti is called a subtree.

nature of the tree : recursive hierarchy

Basic Terminology : Tree nodes: Data elements and several branches pointing to their subtrees

Leaf (terminal node): nodes with zero degrees (number of branches)

Branch nodes (non-terminal nodes): nodes with degrees greater than 0

Degree of node: number of Branches

The degree of the tree: the maximum value of the degree of all nodes in the tree

Depth (height) of the tree: the largest layer of leaf nodes in the tree

Child (Knot): The root of the knot point tree is called the child of the node

Ancestor (node): All nodes on the branch from the root to the node.

Descendants (nodes): Any node of a subtree with a node as its root

Brother (Knot): Children of the same parents are called brothers.

Cousin (knot): Parents in the same layer of the knot of each other called cousins

Ordered tree: The subtree of the nodes in the tree is sequentially (not interchangeable) from left to right.

Unordered tree: The subtree of the nodes in the tree is left-to-right without order (can be interchanged).

Forest: M (m≥0) a collection of disjoint trees

two fork tree recursive definition : A finite set of nodes of N (n>=0), or an empty one, or a root node and two disjoint left and right subtrees, and a tree of about two forks.

Five different forms of binary tree:

The nature of the binary tree: (Proof process slightly)

Property 1: There are at most 2i-1 nodes (I>=1) on the first layer of a binary tree.

Property 2: A two-tree with a depth of K has a maximum of 2k-1 nodes (k>=1).

Property 3: To any binary tree, if its terminal node is n0, the degree of 2 of the node is N2, then n0=n2+1.

Property 4: Full binary tree with n nodes with a depth of log2n +1

full two fork Tree : A two-prong tree with a depth of K and a 2k-1 node.

A complete binary tree : If a depth is k, the nodes in the two-tree with n nodes can correspond to nodes of the full two-tree numbering from 1 to n in sequential numbers with a depth of K. All leaf nodes appear on the K-level or k-1 layer. If the maximum level of the right subtree of any node is L, the maximum level of the left subtree is 1 or l+1.

Binary Tree Storage structure: (1) sequential storage structure

(2) Chain-type storage structure

The following code implements a chain-store structure

1#include <stdio.h>2#include <stdlib.h>3     4 #defineOK 15 #defineERROR-16 #defineOVERFLOW-27  8         9typedefintTElem;TentypedefintStatus; One      AtypedefstructBitnode - { - TElem data; the     structBitnode *lchild,*rchild;//child hands left and right -} bitnode,*Bitree; -  - /** * Function prototype **/ +Status Createbitree (Bitree &T); - //Ansion sequence Input binary tree node value, empty characters represents an empty tree, constructs a two fork tree table representation of the two-fork tree T +Status Preordertraverse (Bitree t,status (*visit) (TElem e)); A //using binary tree table storage structure, visit is the application function of corresponding node operation. at //first-order traversal of binary tree T, call function visit once and only once for each node. - //once visit () fails, the operation fails -Status Inordertraverse (Bitree t,status (*visit) (TElem e)); - //the middle sequence traverses the binary tree T, calling the function visit once and only once for each node. -Status Postordertraverse (Bitree t,status (*visit) (TElem e)); - //Post -order traversal of the binary tree T, call function for each node visit once and only once in voidvisit (TElem e); - //Output to   + voidvisit (TElem e) - { theprintf"%d", E);  * }  $ Panax NotoginsengStatus Createbitree (Bitree *T) - { the TElem E; +scanf"%d",&e); A      the     if(e==0) +     { -*t=NULL; $     } $      -     Else -         { the*t = (bitree) malloc (sizeof(Bitnode)); -                 if(!T)Wuyi                 { the exit (OVERFLOW); -                  } Wu(*t)->data =e; -Createbitree (& (*t)->lchild);//Create a left sub-tree AboutCreatebitree (& (*t)->rchild);//Create right subtree $     } -     return 0; - }  -  AStatus Preordertraverse (Bitree T,void(*visit) (TElem))  + { the      -     if(T) $     { theVisit (t->data);  thePreordertraverse (t->lchild,visit); thePreordertraverse (t->rchild,visit); the  -     } in     Else returnOK; the } the  About  theStatus Inordertraverse (Bitree T,void(*visit) (TElem)) the { the     if(T) +     { -Inordertraverse (t->lchild,visit); theVisit (t->data);BayiInordertraverse (t->rchild,visit); the     }     the      - } -  theStatus Postordertraverse (Bitree T,void(*visit) (TElem)) the { the     if(T) the     { -Postordertraverse (t->lchild,visit); thePostordertraverse (t->rchild,visit); theVisit (t->data); the     }94 }  the  the intMain () the {98 Bitree T; About      -printf"Create a tree, enter 0 for the empty number");101Createbitree (&T);102printf"first-order traversal:");103Preordertraverse (T, *visit);104printf"\ n Sequence traversal:"); theInordertraverse (T, *visit);106printf"\ nthe post-traversal:");107Postordertraverse (T, *visit);108printf"\ n"); 109}

Tree and two-fork tree