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Chapter 4. Trees: Part 1

Chapter 4. Trees: Part 1. Section 4.1 Theory and Terminology Preorder, Postorder and Levelorder Traversals. Theory and Terminology. Definition: A tree is a connected graph with no cycles Consequences: Between any two vertices, there is exactly one unique path. A Tree? No. 9. 10.

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Chapter 4. Trees: Part 1

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  1. Chapter 4. Trees: Part 1 • Section 4.1 • Theory and Terminology • Preorder, Postorder and Levelorder Traversals

  2. Theory and Terminology • Definition: A tree is a connected graph with no cycles • Consequences: • Between any two vertices, there is exactly one unique path

  3. A Tree? No 9 10 11 12 5 6 7 8 2 3 4 1

  4. A Tree? Yes 9 10 11 12 5 6 7 8 2 3 4 1

  5. Theory and Terminology 1 root Edge 2 3 4 Parent of 5, 6 Child of 1, sibling of 2, 3 5 6 7 8 Leaf Height = 1, depth = 2 9 10 11 12 Descendent of 1, 2, 6 Vertex/node Leaf Height of the tree is the height of the root

  6. Preorder Traversal • Visit vertex, then visit child vertices (recursive definition) • How to implement preorder traversal • Depth-first search • Begin at root • Visit vertex on arrival • Implementation may be recursive, stack-based, or nested loop

  7. Preorder Traversal 1 root 1 root (1) (2) 2 3 4 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root 2 3 4 (3) 2 3 4 (4) 5 6 7 8 5 6 7 8

  8. Preorder Traversal 1 root 1 root (5) (6) 2 3 4 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root (7) 2 3 4 2 3 4 (8) 5 6 7 8 5 6 7 8

  9. Postorder Traversal • Visit child vertices, then visit vertex (recursive definition) • How to implement the postorder traversal • Depth-first search • Begin at root • Visit vertex on departure • Implementation may be recursive, stack-based, or nested loop

  10. Postorder Traversal 1 root 1 root (1) 2 3 4 (2) 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root 2 3 4 (3) 2 3 4 (4) 5 6 7 8 5 6 7 8

  11. Postorder Traversal 1 root 1 root (5) (6) 2 3 4 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root (7) 2 3 4 (8) 2 3 4 5 6 7 8 5 6 7 8

  12. Postorder Traversal 1 root 1 root (9) 2 3 4 (10) 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root 2 3 4 (11) (12) 2 3 4 5 6 7 8 5 6 7 8

  13. Postorder Traversal 1 root 1 root (13) 2 3 4 (14) 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root 2 3 4 2 3 4 (15) (16) 5 6 7 8 5 6 7 8

  14. Level-order Traversal • Visit all vertices in level, starting with level 0 and increasing • How to implement levelorder traversal • Breadth-first search • Begin at root • Visit vertex on departure • Only practical implementation is queue-based

  15. Levelorder Traversal 1 root 1 root (1) 2 3 4 (2) 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root (3) 2 3 4 2 3 4 (4) 5 6 7 8 5 6 7 8

  16. Levelorder Traversal 1 root 1 root (5) (6) 2 3 4 2 3 4 5 6 7 8 5 6 7 8 1 root 1 root (7) 2 3 4 2 3 4 (8) 5 6 7 8 5 6 7 8

  17. Traversal Orderings • Preorder: depth-first search (recursion or stack-based), visit on arrival • Postorder: depth-first search (recursion or stack-based), visit on departure • Levelorder: breadth-first search (queue-based), visit on departure

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