1. Traversals • A systematic method to visit all nodes in a tree • Binary tree traversals: • Pre-order: root, left, right • In-order: left, root, right • Post-order: left, right, root • General graph traversals (searches) • Depth-first search • Breadth-first search

2. Inorder(tree t) • if t = nil • return • inorder(t.left) • visit(t) // e.g., print it • inorder(t.right)

3. 5 2 1 3 8 6 10 7 9 Inorder (Infix) 1 2 3 5 6 7 8 9 10 (a BST will always work well with in-order traversal)

4. 5 2 1 3 8 6 10 7 9 Pre-order (Prefix) 5 2 1 3 8 6 7 10 9

5. 5 2 1 3 8 6 10 7 9 Post-order (Postfix) 1 3 2 7 6 9 10 8 5

6. + * 1 3 - 6 10 Post-order 1 3 * + 6 10 -

7. Depth-first search (DFS)חיפוש לעומק)) • DFS: Search one subtree completely before other • Pre-order traversal is an example of a DFS: • Visit root, left subtree (all the way), visit right subtree • We can do it in other order: • Visit root, right subtree, left subtree

8. 5 2 1 3 8 6 10 7 9 Depth-first search (DFS) DFS: visit all descendents before siblings 5 2 1 3 8 6 7 10 9

9. DFS(tree t) • q  new stack • push(q, t) • while (not empty(q)) • curr  pop(q) • visit curr // e.g., print curr.datum • push(q, curr.left) • push(q, curr.right) This version for binary trees only!

10. 5 2 1 3 8 6 10 7 9 Breadth-first search (BFS)(חיפוש לרוחב) BFS: visit all siblings before their descendents 5 2 8 1 3 6 10 7 9

11. BFS(tree t) • q  new queue • enqueue(q, t) • while (not empty(q)) • curr  dequeue(q) • visit curr // e.g., print curr.datum • enqueue(q, curr.left) • enqueue(q, curr.right) This version for binary trees only!

12. DFS(tree t) • q  new stack • push(q, t) • while (not empty(q)) • curr  pop(q) • visit curr // e.g., print curr.datum • push(q, curr.left) • push(q, curr.right) This version for binary trees only!

13. DFS(tree t) Void Graph::dfs (Vertex v) {v.visted = true; for each w adjacent to v if (!w.visited) dfs(w); } This version for any type of trees (graph)

14. Graphs vs. Trees • Graphs don’t have any root • Graphs can be directed or undirected • Trees only grow down (upside-down) • (Why do trees grow upside down for Computer scientists???) • Graphs can have cycles, trees can’t (why?)

15. DFS Example on Graph sourcevertex

16. AVL (Adelson, Velskii, Landis) • Balance the tree (not our targil) • The left and right branches of the tree can only differ by one level • Ensures log N depth (much better for searching) • Takes log N to add or delete

17. 5 3 8 4 10 1 5 Not AVL Tree 3 4 1 AVL Tree

18. AVL Trees • Trees often need to be “rotated” when deleting or inserting to keep AVL balance • Nice link on this process • Sample AVL code