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Trees

Trees . A Quick Introduction to Graphs Definition of Trees Rooted Trees Binary Trees Binary Search Trees. Introduction to Graphs. A graph is a finite set of nodes with edges between nodes Formally, a graph G is a structure (V,E) consisting of a finite set V called the set of nodes, and

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Trees

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  1. Trees • A Quick Introduction to Graphs • Definition of Trees • Rooted Trees • Binary Trees • Binary Search Trees CS 103

  2. Introduction to Graphs • A graph is a finite set of nodes with edges between nodes • Formally, a graph G is a structure (V,E) consisting of • a finite set V called the set of nodes, and • a set E that is a subset of VxV. That is, E is a set of pairs of the form (x,y) where x and y are nodes in V CS 103

  3. Examples of Graphs • V={1,2,3,4,5} • E={(1,2), (2,3), (2,4), (4,2), (3,3), (5,4)} 2 When (x,y) is an edge, we say that x is adjacent to y. 1 is adjacent to 2. 2 is not adjacent to 1. 4 is not adjacent to 3. 1 3 5 4 CS 103

  4. A “Real-life” Example of a Graph • V=set of 6 people: John, Mary, Joe, Helen, Tom, and Paul, of ages 12, 15, 12, 15, 13, and 13, respectively. • E ={(x,y) | if x is younger than y} Mary Helen Joe John Tom Paul CS 103

  5. The nodes represent entities (such as people, cities, computers, words, etc.) Edges (x,y) represent relationships between entities x and y, such as: “x loves y” “x hates y” “x is as smart as y” “x is a sibling of y” “x is bigger than y” ‘x is faster than y”, … Intuition Behind Graphs CS 103

  6. Directed vs. Undirected Graphs • If the directions of the edges matter, then we show the edge directions, and the graph is called a directed graph (or a digraph) • The previous two examples are digraphs • If the relationships represented by the edges are symmetric (such as (x,y) is edge if and only if x is a sibling of y), then we don’t show the directions of the edges, and the graph is called an undirected graph. CS 103

  7. Examples of Undirected Graphs • V=set of 6 people: John, Mary, Joe, Helen, Tom, and Paul, where the first 4 are siblings, and the last two are siblings • E ={(x,y) | x and y are siblings} Mary Helen Joe John Tom Paul CS 103

  8. Definition of Some Graph Related Concepts (Paths) • A path in a graph G is a sequence of nodes x1, x2, …,xk, such that there is an edge from each node the next one in the sequence • For example, in the first example graph, the sequence 4, 1, 2, 3 is a path, but the sequence 1, 4, 5 is not a path because (1,4) is not an edge • In the “sibling-of” graph, the sequence John, Mary, Joe, Helen is a path, but the sequence Helen, Tom, Paul is not a path CS 103

  9. Definition of Some Graph Related Concepts (Cycles) • A cycle in a graph G is a path where the last node is the same as the first node. • In the “sibling-of” graph, the sequence John, Mary, Joe, Helen, John is a cycle, but the sequence Helen, Tom, Paul, Helen is not a cycle CS 103

  10. Graph Connectivity • An undirected graph is said to be connected if there is a path between every pair of nodes. Otherwise, the graph is disconnected • Informally, an undirected graph is connected if it hangs in one piece Connected Disconnected CS 103

  11. Graph Cyclicity • An undirected graph is cyclic if it has at least one cycle. Otherwise, it is acyclic Connected and cyclic Disconnected and cyclic Connected and acyclic Disconnected and acyclic CS 103

  12. Trees • A tree is a connected acyclic undirected graph. The following are three trees: 8 9 2 10 1 7 5 12 3 11 4 6 CS 103

  13. Rooted Trees • A rooted tree is a tree where one of the nodes is designated as the root node. (Only one root in a tree) • A rooted tree has a hierarchical structure: the root on top, followed by the nodes adjacent to it right below, followed by the nodes adjacent to those next, and so on. CS 103

  14. 1 3 2 7 5 8 9 10 4 6 11 12 Example of a Rooted Tree 8 9 2 10 1 7 5 12 3 11 4 6 Unrooted tree Tree rooted with root 1 CS 103

  15. 1 3 2 7 5 8 9 10 4 6 11 12 Tree-Related Concepts • The nodes adjacent to x and below x are called the children of x, and x is called their parents • A node that has no children is called a leaf • The descendents of a node are: itself, its children, their children, all the way down • The ancestors of a node are: itself, its parent, its grandparent, all the way to the root CS 103

  16. 1 3 2 7 5 8 9 10 4 6 11 12 Tree-Related Concepts (Contd.) • The depth of a node is the number of edges from the root to that node. • The depth (or height) of a rooted tree is the depth of the lowest leaf • Depth of node 10: 3 • Depth of this tree: 4 CS 103

  17. 1 3 2 7 5 8 9 10 4 6 11 12 Binary Trees • A tree is a binary tree if every node has at most two children 1 3 7 5 8 9 10 4 6 11 12 Nonbinary tree Binary tree CS 103

  18. 1 3 7 5 8 9 4 6 11 12 Binary-Tree Related Definitions • The children of any node in a binary tree are ordered into a left child and a right child • A node can have a left and a right child, a left child only, a right child only, or no children • The tree made up of a left child (of a node x) and all its descendents is called the left subtree of x • Right subtrees are defined similarly 10 CS 103

  19. Graphical View Binary-tree Nodes Graphically, a TreeNode is: • A binary-tree node • consists of 3 parts: • Data • Pointer to left child • Pointer to right child data left right In practice, a TreeNode will be shown as a circle where the data is put inside, and the node label (if any) is put outside. 5.8 2 data label CS 103

  20. A Binary-tree Node Class class TreeNode { public: typedef int datatype; TreeNode(datatype x=0, TreeNode *left=NULL, TreeNode *right=NULL){ data=x; this->left=left; this->right=right; }; datatype getData( ) {return data;}; TreeNode *getLeft( ) {returnleft;}; TreeNode *getRight( ) {returnright;}; void setData(datatype x) {data=x;}; void setLeft(TreeNode *ptr) {left=ptr;}; void setRight(TreeNode *ptr) {right=ptr;}; private: datatype data; // different data type for other apps TreeNode *left; // the pointer to left child TreeNode *right; // the pointer to right child }; CS 103

  21. Binary Tree Class class Tree { public: typedef int datatype; Tree(TreeNode *rootPtr=NULL){this->rootPtr=rootPtr;}; TreeNode *search(datatype x); bool insert(datatype x); TreeNode * remove(datatype x); TreeNode *getRoot(){return rootPtr;}; Tree *getLeftSubtree(); Tree *getRightSubtree(); bool isEmpty(){return rootPtr == NULL;}; private: TreeNode *rootPtr; }; CS 103

  22. Binary Search Trees • A binary search tree (BST) is a binary tree where • Every node holds a data value (called key) • For any node x, all the keys in the left subtree of x are ≤ the key of x • For any node x, all the keys in the right subtree of x are > the key of x CS 103

  23. 15 20 8 2 11 27 30 6 22 10 12 3 7 14 Example of a BST CS 103

  24. 15 20 8 27 11 2 30 6 22 10 12 3 7 14 Searching in a BST • To search for a number b: • Compare b with the root; • If b=root, return • If b<root, go left • If b>root, go right • Repeat step 1, comparing b with the new node we are at. • Repeat until either the node is found or we reach a non-existing node • Try it with b=12, and also with b=17 CS 103

  25. Code for Search in BST // returns a pointer to the TreeNode that contains x, // if one is found. Otherwise, it returns NULL TreeNode * Tree::search(datatype x){ if (isEmpty()) {return NULL;} TreeNode *p=rootPtr; while (p != NULL){ datatype a = p->getData(); if (a == x) return p; elseif (x<a) p=p->getLeft(); else p=p->getRight(); } returnNULL; }; CS 103

  26. Insertion into a BST Insert(datatype b, Tree T): • Search for the position of b as if it were in the tree. The position is the left or right child of some node x. • Create a new node, and assign its address to the appropriate pointer field in x • Assign b to the data field of the new node CS 103

  27. 15 20 8 2 11 27 30 22 6 10 12 3 7 14 Illustration of Insert 15 20 8 2 11 27 30 22 6 10 12 25 3 7 14 Before inserting 25 After inserting 25 CS 103

  28. Code for Insert in BST bool Tree::insert(datatype x){ if (isEmpty()) {rootPtr = new TreeNode(x);return true; } TreeNode *p=rootPtr; while (p != NULL){ datatype a = p->getData(); if (a == x) return false; // data is already there else if (x<a){ if (p->getLeft() == NULL){ // place to insert TreeNode *newNodePtr= new TreeNode(x); p->setLeft(newNodePtr); return true;} else p=p->getLeft(); }else { // a>a if (p->getRight() == NULL){ // place to insert TreeNode *newNodePtr= new TreeNode(x); p->setRight(newNodePtr); return true;} else p=p->getRight();} } }; CS 103

  29. Deletion from a BST • Illustration in class CS 103

  30. Deletion from a BST (pseudocode) Delete(datatype b, Tree T) • Search for b in tree T. If not found, return. • Call x the first node found to contain b • If x is a leaf, remove x and set the appropriate pointer in the parent of x to NULL • If x has only one child y, remove x, and the parent of x become a direct parent of y (More on the next slide) CS 103

  31. Deletion (contd.) 5. If x has two children, go to the left subtree, and find there in largest node, and call it y. The node y can be found by tracing the rightmost path until the end. Note that y is either a leaf or has no right child 6. Copy the data field of y onto the data field of x 7. Now delete node y in a manner similar to step 4. CS 103

  32. Code for Delete in BST(4 slides) // finds x in the tree, removes it, and returns a pointer to the containing // TreeNode. If x is not found, the function returns NULL. TreeNode * Tree::remove(datatype x){ if (isEmpty()) return NULL; TreeNode *p=rootPtr; TreeNode *parent = NULL; // parent of p char whatChild; // 'L' if p is a left child, 'R' O.W. while (p != NULL){ datatype a = p->getData(); if (a == x) break; // x found else if(x<a) { parent = p; whatChild = 'L'; p=p->getLeft();} else {parent = p; whatChild = 'R'; p=p->getRight();} } if (p==NULL) return NULL; // x was not found CS 103

  33. // Handle the case where p is a leaf. • // Turn the appropriate pointer in its parent to NULL • if (p->getLeft() == NULL && p->getRight() == NULL){ • if (parent != NULL) // x is not at the root • if (whatChild == 'L') • parent->setLeft(NULL); • else • parent->setRight(NULL); • else // x is at the root • rootPtr=NULL; • return p; • } CS 103

  34. else if (p->getLeft() == NULL){ // p has only one a child -- a right child. Let the parent of p // become an immediate parent of the right child of p. if (parent != NULL) // p is not the root if (whatChild == 'L') parent->setLeft(p->getRight()); else parent->setRight(p->getRight()); else rootPtr=p->getRight(); // p is the root return p; } else if (p->getRight() == NULL){ // p has only one a child -- a left child. Let the parent of p // become an immediate parent of the left child of p. if (parent != NULL) // p is not the root if (whatChild == 'L') parent->setLeft(p->getLeft()); else parent->setRight(p->getLeft()); else rootPtr=p->getLeft(); // p is the root return p; } CS 103

  35. else { // p has two children TreeNode *returnNode= new TreeNode(*p); // replicates p TreeNode * leftChild = p->getLeft(); if (leftChild->getRight() == NULL){// leftChild has no right child p->setData(leftChild->getData()); p->setLeft(leftChild->getLeft()); delete leftChild; return returnNode; } TreeNode * maxLeft = leftChild->getRight(); TreeNode * parent2 = leftChild; while (maxLeft != NULL){parent2 = maxLeft; maxLeft = maxLeft ->getRight();} // now maxLeft is the node to swap with p. p->setData(maxLeft->getData()); if (maxLeft->getLeft()==NULL) parent2->setRight(NULL); // maxLeft a leaf else parent2->setRight(maxLeft->getLeft()); //maxLeft not a leaf delete maxLeft; return returnNode; } }; CS 103

  36. Additional Things for YOU to Do • Add a method to the Tree class for returning the maximum value in the BST • Add a method to the Tree class for returning the minimum value in the BST • Write a function that takes as input an array of type datatype, and an integer n representing the length of the array, and returns a BST Tree Object containing the elements of the input array CS 103

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