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CMPT 225

CMPT 225. Binary Search Trees. Trees. A set of nodes with a single starting point called the root Each node is connected by an edge to some other node A tree is a connected graph There is a path to every node in the tree There are no cycles in the tree.

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CMPT 225

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  1. CMPT 225 Binary Search Trees

  2. Trees • A set of nodes with a single starting point • called the root • Each node is connected by an edge to some other node • A tree is a connected graph • There is a path to every node in the tree • There are no cycles in the tree. • It can be proved by MI that a tree has one less edge than the number of nodes. • Usually depicted with the root at the top

  3. Is it a Tree? NO! All the nodes are not connected yes! (but not a binary tree) yes! NO! There is a cycle and an extra edge (5 nodes and 5 edges) yes! (it’s actually the same graph as the blue one) – but usually we draw tree by its “levels”

  4. Examples of trees • directory structure • family trees: • all descendants of a particular person • all ancestors born after year 1800 of a particular person • evolutionary tress (also called phylogenetic trees) • albegraic expressions

  5. Examples of trees Binary trees that represent algebraic expressions

  6. A B C D E F G Tree Relationships • If there is an edge between two nodes u and v, and u is “above” v in the tree (closer to the root), then v is said to be a child of u, and u the parent of v • A is the parent of B, C and D • This relationship can be generalized (transitively) • E and F are descendants of A • D and A are ancestors of G • B, C and D are siblings

  7. More Tree Terminology • A leaf is a node with no children • A path[a branch] is a sequence of nodes v1 … vn • where vi is a parent of vi+1 (1  i  n-1) [and v1 is aroot and vn is a leaf] • A subtree is any node in the tree along with all of its descendants • A binary tree is a tree with at most two children per node • The children are referred to as left and right (i.e., children are usually ordered) • We can also refer to left and right subtrees of a node

  8. C C A F E G G D Tree Terminology Example A path from A to D to G B D subtree rooted at B leaves: C,E,F,G E F G

  9. Binary Tree A B C right child of A left subtree of A F D E G right subtree of C H I J

  10. Measuring Trees • The height of a node v is the number of nodes on the longest path from v to a leaf • The height of the tree is the height of the root, which is the number of nodes on the longest path from the root to a leaf • The depth of a node v is the number of nodes on the path from the root to v • This is also referred to as the level of a node • Note that there are slightly different formulations of the height of a tree • Where the height of a tree is said to be the length (the number of edges) on the longest path from node to a leaf

  11. A Height of a Binary Tree height of the tree is 4 A level 2 height of node B is 3 B B C F E level 3 D E G depth of node E is 3 H I J level 4

  12. Representation of binary trees publicclass TreeNode<T> { private T item; private TreeNode<T> leftChild; private TreeNode<T> rightChild; public TreeNode(T newItem) { // Initializes tree node with item and no children (a leaf). item = newItem; leftChild = null; rightChild = null; } // end constructor public TreeNode(T newItem, TreeNode<T> left, TreeNode<T> right) { // Initializes tree node with item and // the left and right children references. item = newItem; leftChild = left; rightChild = right; } // end constructor public T getItem() { // Returns the item field. return item; } // end getItem

  13. publicvoid setItem(T newItem) { // Sets the item field to the new value newItem. item = newItem; } // end setItem public TreeNode<T> getLeft() { // Returns the reference to the left child. return leftChild; } // end getLeft publicvoid setLeft(TreeNode<T> left) { // Sets the left child reference to left. leftChild = left; } // end setLeft public TreeNode<T> getRight() { // Returns the reference to the right child. return rightChild; } // end getRight publicvoid setRight(TreeNode<T> right) { // Sets the right child reference to right. rightChild = right; } // end setRight } // end TreeNode

  14. 2 5 4 1 Representing a tree TreeNode<Integer> root=new TreeNode<Integer>(new Integer(2)); TreeNode<Integer> root.setLeft( new TreeNode<Integer>(new Integer(5), new TreeNode<Integer>(new Integer(4)), new TreeNode<Integer>(new Integer(1)))); • How to compute height of a node? • if tree is empty, height is 0 (no levels at all) • if tree has just a root, height is 1 • height(T) = 1 + max(height(left-subtree(T),height(right-subtree(T))

  15. Special types of Binary Trees • A binary tree is full if no node has only one child and if all the leaves have the same depth • A full binary tree of height h has (2h – 1) nodes, of which 2h-1 are leaves • A complete binary tree is one where • The leaves are on at most two different levels, • The second to bottom level is filled in and • The leaves on the bottom level are as far to the left as possible. • A balanced binary tree is one where • No leaf is more than a certain amount farther from the root than any other leaf, this is sometimes stated more specifically as: • The height of any node’s right subtree is at most one different from the height of its left subtree • A balanced tree’s height is sometime specified in terms of its relation to the number of nodes in the tree (see red-black trees)

  16. A A B C B C D E F F D E G Perfect and Complete Binary Trees Complete binary tree Full binary tree

  17. A A B C B C E D F D E F G Balanced Binary Trees

  18. A B C A E D B F C D Unbalanced Binary Trees

  19. Binary Tree Traversals • A traversal algorithm for a binary tree visits each node in the tree • and, typically, does something while visiting each node! • Traversal algorithms are naturally recursive • There are three traversal methods • Inorder • Preorder • Postorder

  20. InOrder Traversal Algorithm // InOrder traversal algorithm inOrder(TreeNode<T> n) { if (n != null) { inOrder(n.getLeft()); visit(n) inOrder(n.getRight()); } }

  21. PreOrder Traversal visit(n) 1 preOrder(n.leftChild) 17 preOrder(n.rightChild) visit preOrder(l) preOrder(r) visit preOrder(l) preOrder(r) 2 13 6 27 3 9 visit preOrder(l) preOrder(r) 5 16 7 20 8 39 visit preOrder(l) preOrder(r) visit preOrder(l) preOrder(r) visit preOrder(l) preOrder(r) 4 11 visit preOrder(l) preOrder(r)

  22. PostOrder Traversal postOrder(n.leftChild) 8 postOrder(n.rightChild) 17 visit(n) postOrder(l) postOrder(r) visit postOrder(l) postOrder(r) visit 4 13 7 27 2 9 postOrder(l) postOrder(r) visit 3 16 5 20 6 39 postOrder(l) postOrder(r) visit postOrder(l) postOrder(r) visit postOrder(l) postOrder(r) visit 1 11 postOrder(l) postOrder(r) visit

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