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Trees

Trees. Chapter 9. Tree. graph connected undirected no simple circuits (acyclic) no multiple edges no loops. Sample Trees?. Tree Tree Not Not. Theorem 1. An undirected graph is a tree iff A simple path exists in a tree between any two vertices. Root.

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Trees

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  1. Trees Chapter 9

  2. Tree • graph • connected • undirected • no simple circuits (acyclic) • no multiple edges • no loops

  3. Sample Trees? Tree Tree Not Not

  4. Theorem 1 • An undirected graph is a tree iff • A simple path exists in a tree • between any two vertices

  5. Root • A particular tree vertex • from which we assign a direction to each edge • Each edge is directed away from the root

  6. Rooted tree • A tree with a designated root • A directed graph • Direction of all edges is away from root

  7. Parent • In a rooted tree, • a parent of vertex v is • the unique vertex u • such that there is a directed edge from u to v

  8. Child • The vertex v • to which a directed edge exists • from parent u in a rooted tree

  9. Siblings • vertices with the same parent

  10. Leaf • a vertex of a tree that has no children

  11. Ancestors of node A • nodes located on the path • from A to the root

  12. Descendants of node A • nodes located on the path • from A to a leaf node

  13. Internal vertices • Vertices with children

  14. Sub-tree • a tree • contained in a larger tree • whose root may be a child node • in the larger tree

  15. m-ary tree • a rooted tree • with no more than m children per vertex

  16. Full m-ary tree • a rooted tree • whose every internal vertex • has exactly m children

  17. Theorem 2 • A tree with n vertices has n - 1 edges. 7 vertices 6 edges

  18. Theorem 3 • A full m-ary tree • with i internal vertices • contains n = mi + 1 vertices. m = 2 i = 7 15 vertices

  19. Tree Height • height (level) of a node • the length of the path from the root to a node • height of a tree • the length of the longest path in a tree

  20. The maximum number of nodes at any level is mh • h is height of a node at that level of the tree 212223

  21. The minimum number of nodes • of a tree of height h is • h+1

  22. The maximum number of nodes • in a tree of height h is • m(h+1) -1 • 2(3+1) - 1

  23. Balanced tree • A rooted m-ary tree of height h • is called balanced • if all leaves are at level h or h - 1 YES NO YES

  24. If an m-ary tree of height h • has l leaves, • and the tree is full and balanced, • h = ceil(log m l) • h = ceil (log28) • h = 3 What does this imply about access speed if a tree is used as a data structure?

  25. Applications of Trees 8.2

  26. Binary search tree • A binary tree where key value in any node is • greater than key of its left child • and any of its children • (the nodes in the left subtree) • less than key of its right child • and any of its children • (the nodes in the right subtree) • http://math.nemcc.edu/bst/

  27. Binary Search Tree Example

  28. Form a BST with the words Mathematics, Physics, Geography, Zoology, Meteorology, Geology, Psychology, Chemistry

  29. NOTE:Input order determines a tree's shape.Tree Animation

  30. Tree Traversal 8.3

  31. Inorder Tree Traversal • process Left subtree inorder • Visit a node (or process node) • Process Right subtree inorder • Processes BST vertices in ascending sequence • http://nova.umuc.edu/~jarc/idsv/lesson1.html

  32. Inorder Traversal Example LVR Arps, Dietz, Egofske, Fairchild, Garth, Huston, Keith Magillicuddy, Nathan, Perkins, Seliger, Talbot, Underwood,Verkins, Zarda

  33. Preorder Tree Traversal • allows quickest access to the whole tree • VISIT a node • process LEFT subtree in preorder • process RIGHT subtree in preorder

  34. Preorder Traversal Example VLR Magillicuddy, Fairchild, Dietz, Arps, Egofske, Huston, Garth, Keith, Talbot, Perkins, Nathan, Selinger, Verkins, Underwood, Zarda

  35. Postorder Tree Traversal • good for deletion of nodes; postfix notation • process LEFT subtree in postorder • process RIGHT subtree in postorder • VISIT a node

  36. Postorder Traversal Example LRV Arps, Egofske, Dietz, Garth, Keith, Huston, Fairchild, Nathan, Selinger, Perkinds, Underwood, Zarda, Verkins, Talbot, Magillicuddy

  37. Expression Tree • An ordered rooted tree • associates operands & operators in a uniform way +

  38. Give Pre, In, Postorder • PreOrder: • - + + * 6 2 7 * 8 3 / 6 7 • InOrder: • 6 * 2 + 7 + 8 * 3 - 6 / 7 • PostOrder: • 6 2 * 7 + 8 3 * + 6 7 / - +

  39. Spanning Trees 9.4

  40. Spanning Subgraph • A spanning subgraph of G is • G’ = (V, E’) • where E’ is a subset of E • Note every vertex of G is included

  41. Spanning Tree • A spanning subgraph that is a tree • connected • acyclic • See p. 581-2

  42. Depth First Search • A procedure for constructing a spanning tree • by adding edges that form a path until this is not possible • then moving back up the tree • until a vertex is found where a new path can be formed • http://www.cs.sunysb.edu/~skiena/combinatorica/animations/search.html

  43. Breadth First Search • A procedure for constructing a spanning tree • that successively adds all edges incident to the last set of edges added • unless a simple circuit isformed • http://www.cs.duke.edu/~wcp/DFSanim.html • http://152.3.140.5/~wcp/DFSanim.html

  44. Perform DFS, BFS search DFS: a, b, c, d, e, f, g BFS: a b c g d e f

  45. Perform DFS, BFS search DFS: a, b, c, d, e, f BFS: a b d e c f

  46. Minimum Spanning Tree • A connected weighted graph • is a spanning tree that has • the smallest possible sum of weights of its edges. • http://study.haifa.ac.il/~hvaiderm/sem3.html

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