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More on Trees

More on Trees. University. Fac. of Sci. & Eng. Law School. Bus. School. Math. Dept. CS Dept. EE Dept. . Linked Structure for Trees. A node is represented by an object storing Element Parent node Sequence of children nodes Node objects implement the Position ADT. B. . . A. D.

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More on Trees

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  1. More onTrees University Fac. of Sci. & Eng. Law School Bus. School Math. Dept. CS Dept. EE Dept. Trees

  2. Linked Structure for Trees • A node is represented by an object storing • Element • Parent node • Sequence of children nodes • Node objects implement the Position ADT B   A D F B A D F   C E C E Trees

  3. D C A B E Linked Structure for Binary Trees  • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT   B A D     C E Trees

  4. D C B A E Linked Structure for Binary Trees  • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT   B A D     D C     F Trees G G F

  5. Preorder Traversal AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) • A traversal visits the nodes of a tree in a systematic manner • In a preorder traversal, a node is visited before its descendants • Application: print a structured document 1 Make Money Fast! 2 5 9 1. Motivations 2. Methods References 6 7 8 3 4 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity Trees

  6. Preorder Traversal (Another example) AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) My Explanation: • When a node is reached, visit it. • Visit the sub-trees rooted by its children one by one. 1 2 17 9 14 3 6 10 13 4 5 7 Trees 11 12 16 8 15

  7. Postorder Traversal AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) • In a postorder traversal, a node is visited after its descendants • Application: compute space used by files in a directory and its subdirectories 9 cs16/ 8 3 7 todo.txt1K homeworks/ programs/ 4 5 6 1 2 Robot.java20K h1c.doc3K h1nc.doc2K DDR.java10K Stocks.java25K Trees

  8. Postorder Traversal (Another example) My explanation: • If the reached node is a leaf, then visit it. • When a node is visited, visit the sub-tree rooted by its sibling on the right. • When the leftmost child is visited, visit its parent. AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) . 17 7 16 15 14 3 6 10 11 1 2 4 Trees 8 9 13 5 12

  9. +   2 - 3 2 5 1 Evaluate Arithmetic Expressions AlgorithmevalExpr(v) ifisExternal (v) returnv.element () else x evalExpr(leftChild (v)) y evalExpr(rightChild (v))  operator stored at v returnx  y • Specialization of a postorder traversal • recursive method returning the value of a subtree • when visiting an internal node, combine the values of the subtrees Trees

  10. A … B D C E F J G H Array-Based Representation of Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11 Trees

  11. Full Binary Tree • A full binary tree: • All the leaves are at the bottom level • All nodes which are not at the bottom level have two children. • A full binary of height h has 2h leaves and 2h-1 internal nodes. 1 3 2 4 6 7 5 This is not a full binary tree. A full binary tree of height 2 Trees

  12. Properties of Proper Binary Trees • Properties for proper binary tree: • e = i +1 • n =2e -1 • h  i • e 2h • h log2e • Notation n number of nodes e number of external nodes i number of internal nodes h height 1 3 2 7 6 No need to remember. 14 15 Trees

  13. Depth(v): no. of ancestors of v Algorithmdepth(T,v) If T.isRoot(v) then return 0; else return 1+depth(T, T.parent(v)) 0 Make Money Fast! 1 1 1 1. Motivations 2. Methods References 2 2 2 2 2 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity Trees

  14. Height(T): the height of T is max depth of an external node Algorithm height1(T) h=0; for each vT.positions() do if T.isExternal(v) then h=max(h, depth(T, v)) return h T.positions() holds all nodes in T. See the method in LinkedBinaryTree.java. The max is 2. 0 Make Money Fast! 1 1 1. Motivations 2. Methods 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 2 2 2 2 1.1 Greed 1.2 Avidity Trees 2

  15. Height(T,v): • If v is an external node, then height of v is 0. • Otherwise, the height of v is one +max height of a child of v. Algorithm height2(T,v) if T.isExternal(v) then return 0 else h=0 for each wT.children(v) do h=max(h, height2(T, w)) return 1+h Trees

  16. Height(T,v): Algorithm height2(T,v) if T.isExternal(v) then return 0 else h=0 for each wT.children(v) do h=max(h, height2(T, w)) return 1+h 2 Make Money Fast! 1 1 1 1. Motivations 2. Methods References 0 0 0 0 0 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity Trees

  17. +   2 - 3 2 5 1 Next time: Chapter 7. Heap Exercise1: Given a binary tree T, and two nodes u and v in T, test if u is an ancestor of v. Boolean isAncestor(T, u, v) Exercise2: Suppose that we use array-based representation for a binary tree. Give the positions of the nodes in the following tree This time is slightly worse than last time. I should prepare the speech carefully. 5.33? Trees

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