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Trees

Trees The definitions for this presentation are from from: Corman , et. al., Introduction to Algorithms (MIT Press), Chapter 5. Some material on binomial trees is from Hull. A Few Applications. Arithmetic Expressions b + a * b. +. b. *. b. a. A Few Applications.

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Trees

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  1. Trees The definitions for this presentation are from from: Corman , et. al., Introduction to Algorithms (MIT Press), Chapter 5. Some material on binomial trees is from Hull. Data Structures and Algorithms

  2. A Few Applications Arithmetic Expressions b + a * b + b * b a Data Structures and Algorithms

  3. A Few Applications <employee> XML Document Object Model <ssn> title <name> #text #text #text Data Structures and Algorithms

  4. A Few Applications Binomial “Trees” Movements in Time dt Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d. Su p S 1 – p Sd From Hull

  5. Definitions A Free tree is a connected, acyclic undirected graph. Data Structures and Algorithms

  6. If an undirected graph is acyclic but possibly disconnected, it is a forest. Data Structures and Algorithms

  7. This is a graph that contains a cycle and is therefore neither a tree nor a forest. Data Structures and Algorithms

  8. Theorem (Properties of Free Trees) Let G = (V, E) be an undirected graph. The following statements are equivalent: 1. G is a free tree. Data Structures and Algorithms

  9. 2. Any two vertices in G are connected by a unique simple path. Data Structures and Algorithms

  10. 3. G is connected, but if any edge is removed from E, the resulting graph is disconnected. Data Structures and Algorithms

  11. 4. G is connected, and |E| = |V| - 1. Data Structures and Algorithms

  12. 5. G is acyclic, and |E| = |V| - 1. Data Structures and Algorithms

  13. 6. G is acyclic, but if any edge is added to E, the resulting graph contains a cycle. Data Structures and Algorithms

  14. A rooted tree is a free tree in which one of the vertices is distinguished from the others. The distinguished vertex is called the root of the tree. We often refer to a vertex of a rooted tree as a node (we may also call this a vertex) of the tree. The following figure shows a rooted tree on a set of 12 nodes with root 7. 7 3 10 4 8 12 11 2 6 5 1 9 Data Structures and Algorithms

  15. Consider a node x in a rooted tree T with root r. Any node y on the unique path from r to x is called an ancestor of x. If y is an ancestor of x, then x is a descendant of y. (Every node is both an ancestor and a descendant of itself.) If y is an ancestor of x and x  y, then y is a proper ancestor of x and x is a proper descendant of y. The subtree rooted at x is the tree induced by descendantsof x, rooted at x. r 8 y 5 6 x 9 Data Structures and Algorithms

  16. If the last edge on the path from the root r of a tree T to a node x is (y, x), then y is the parent of x, and x is a child of y. The root is the only node in T with no parent. If two nodes have the same parent, they are siblings. A node with no children is an external node or leaf. A nonleaf node is an internal node. Data Structures and Algorithms

  17. The number of children of a node x in a rooted tree T is called the degree of x. The length of the path from the root r to a node x is the depth of x in T. The largest depth of any node in T is the height of T. 7 depth 0 depth 1 3 10 4 height = 4 depth 2 8 12 11 2 6 5 1 depth 3 9 depth 4 Data Structures and Algorithms

  18. An ordered tree is a rooted tree in which the children of each node are ordered. That is, if a node has k children, then there is a first child, a second child, …, and a kth child. The two trees shown below are different when considered to be ordered trees, but the same when considered to be just rooted trees. 7 7 4 3 10 4 3 10 12 8 11 2 11 2 8 12 1 6 5 6 5 1 9 9 Data Structures and Algorithms

  19. A binary tree T is a structure defined on a finite set of • nodes that either • contains no nodes, or • is comprised of three disjoint sets of nodes: • a root node, a binary tree called its left subtree • and a binary tree called its right subtree. 3 2 7 4 1 5 6 Data Structures and Algorithms

  20. Full binary tree: each node is either a leaf or has degree exactly 2. In a positional tree, the children of a node are labeled with distinct positive integers. The ith child of a node is absent if no child is labeled with integer i. A k-ary tree is a positional tree in which for every node, all children with labels greater than k are missing. Thus, a binary tree is a k-ary tree with k = 2. Data Structures and Algorithms

  21. A complete k-ary tree is a k-ary tree in which all leaves have the same depth and all internal nodes have degree k. depth 0 depth 1 depth 2 height = 3 depth 3 Data Structures and Algorithms

  22. How many leaves L does a complete binary tree of height h have? d = 0 d = 1 d = 2 The number of leaves at depth d = 2d If the height of the tree is h it has 2h leaves. L = 2h. Data Structures and Algorithms

  23. What is the height h of a complete binary tree with L leaves? leaves = 1 height = 0 leaves = 2 height = 1 leaves = 4 height = 2 leaves = L height = Log2L Since L = 2h Log2L = Log22h h = Log2L Data Structures and Algorithms

  24. The number of internal nodes of a complete binary tree of height h is ? Internal nodes = 0 height = 0 Internal nodes = 1 height = 1 Internal nodes = 1 + 2 height = 2 Internal nodes = 1 + 2 + 4 height = 3 Geometric series 1 + 2 + 22 + . . . + 2 h-1 =  2i = 2h - 1 2 - 1 Thus, a complete binary tree of height = h has 2h-1 internal nodes. Data Structures and Algorithms

  25. The number of nodes n of a complete binary tree of height h is ? nodes = 1 height = 0 nodes = 3 height = 1 nodes = 7 height = 2 nodes = 2h+1- 1 height = h Since L = 2h and since the number of internal nodes = 2h-1 the total number of nodes n = 2h+2h-1 = 2(2h) – 1 = 2h+1- 1. Data Structures and Algorithms

  26. If the number of nodes is n then what is the height? nodes = 1 height = 0 nodes = 3 height = 1 nodes = 7 height = 2 nodes = n height = Log2(n+1) - 1 Since n = 2h+1-1 n + 1 = 2h+1 Log2(n+1) = Log2 2h+1 Log2(n+1) = h+1 h = Log2(n+1) - 1 Data Structures and Algorithms

  27. Catalan Numbers = 1 (2n)! (n+1) n! (2n-n)! 1,1,2,5,14,... Data Structures and Algorithms

  28. The number of distinct binary trees with n nodes N=3 N = 1 N=2 N = 0 ... Data Structures and Algorithms

  29. Class for Binary Nodes public class BTNode { private Object data; private BTNode left; private BTNode right; ... Data Structures and Algorithms

  30. public BTNode(Object obj, BTNode l, BTNode r) { data = obj; left = l; right= r; } public boolean isLeaf() { return (left == null) && (right == null); } ... Data Structures and Algorithms

  31. Copying Trees public static BTNode treeCopy(BTNode t) { if (t == null) return null; else { BTNode leftCopy = treeCopy(t.left); BTNode rightCopy = treeCopy(t.right); return new BTNode(t.data, leftCopy, rightCopy); } } Data Structures and Algorithms

  32. Tree Traversals • Preorder • Inorder • Postorder • Levelorder Data Structures and Algorithms

  33. public void preOrderPrint(){ System.out.println(data); if (left != null) left.preOrderPrint(); if (right != null) right.preOrderPrint(); } a b c d e f g a e b g c d f Root, Left, Right Data Structures and Algorithms

  34. public void inOrderPrint(){ if (left != null) { left.inOrderPrint() System.out.println(data); if (right != null) right.inOrderPrint() } c b d a f e g a b e c d f g Left, Root, Right Data Structures and Algorithms

  35. public void postOrder(){ if (left != null left.postOrder() if (right != null) right.postOrder() System.out.println(data); } c d b f g e a a b e c d f g Left, right, root Data Structures and Algorithms

  36. levelorder (T) { Q = makeEmptyQueue() enqueue (T,Q) until isempty (Q) { p = dequeue(Q) visit (p) for each child  of P, in order, do enqueue (, Q) } } a b e c d f g a b e c d f g Data Structures and Algorithms

  37. An Array Representation Suppose we are lucky enough to be working with complete binary trees. We can store the tree in an array. Let the root be at index 0 and let the left and right children of node i be at indexes 2i+1 and 2i+2 respectively. Lewis and Denemberg, Page 111 Data Structures and Algorithms

  38. An Array Representation isLeaf(i) : 2i + 1 >= n leftChild(i) : 2i + 1 (none if 2i+ 1 >= n) rightChild(i): 2i + 2 (none if 2i + 2 >= n) leftSibling(i): i - 1 (none if i == 0 or i is odd) rightSibling(i) : i + 1 (none if i = n-1 or i is even) parent(i) = int((i-1)/2) (none if i == 0) Works if the tree is “almost complete”, growing top to bottom and left to right. Lewis and Denemberg, Page 111 Data Structures and Algorithms

  39. Binomial “Trees” Using an Array Representation S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 The array element bt[0] will be S0. In general, given a node with index i at depth d, its left child is located at bt[i + d+1] and its right child is located at bt[i + d+2] S0d4 Data Structures and Algorithms

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