Trees
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This document discusses trees and tree traversal algorithms. It defines trees and forests, and introduces spanning trees and spanning forests. It then discusses ordered rooted trees and the three tree traversal algorithms: preorder, inorder, and postorder. Preorder traverses the root first, then left subtree, then right subtree. Inorder traverses left subtree, then root, then right subtree. Postorder traverses left subtree, then right subtree, then root. Examples of each traversal type on sample trees are provided.
![Inorder Tree Walk
• What does the following code do?
TreeWalk(x)
TreeWalk(left[x]);
print(x);
TreeWalk(right[x]);
• A: prints elements in sorted (increasing) order
• This is called an inorder tree walk
– Preorder tree walk: print root, then left, then right
– Postorder tree walk: print left, then right, then root](https://image.slidesharecdn.com/trees-150209135744-conversion-gate02/85/Trees-7-320.jpg)
Trees
- 2. Trees and Forests • A (free) tree is an undirected graph T such that – T is connected – T has no cycles (circuits) This definition of tree is different from the one of a rooted tree • A rooted tree is a tree in which one vertex has been designated as the root and every edge is directed away from the root. • A forest is an undirected graph without cycles • The connected components of a forest are trees Tree Forest
- 3. Spanning Trees and Forests • A spanning tree of a connected graph is a spanning subgraph that is a tree • A spanning tree is not unique unless the graph is a tree • Spanning trees have applications to the design of communication networks • A spanning forest of a graph is a spanning subgraph that is a forest Graph Spanning tree
- 4. Tree Traversal • Ordered Rooted Tree: An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered. • Tree Traversal: Ordered rooted trees are often used to store information. Tree traversal is the procedure of visiting different vertices of the tree to read information stored in that vertex. There are three different orders of the tree traversal.
- 5. Inorder Traversal • Let T be an ordered rooted tree with root r and T1, T2, …Tn are the subtrees at r from left to right. The in order traversal begins by traversing T1 in inorder, then visiting r. It continues by traversing T2 in inorder, then T3 in inorder, . . . , and finally Tn in inorder.
- 6. Inorder Tree Walk • Left, root, right • Example: • A, B, D, F, H, K F B H KDA
- 7. Inorder Tree Walk • What does the following code do? TreeWalk(x) TreeWalk(left[x]); print(x); TreeWalk(right[x]); • A: prints elements in sorted (increasing) order • This is called an inorder tree walk – Preorder tree walk: print root, then left, then right – Postorder tree walk: print left, then right, then root
- 8. Example – Inorder Traversal • Example 3, p-714
- 9. Preorder Traversal • Let T be an ordered rooted tree with root r and T1, T2, …Tn are the subtrees at r from left to right. The preorder traversal begins by visiting r . It continues by traversing T1 in preorder, then T2 in preorder, then T3 in preorder, . . . , and finally Tn in preorder.
- 10. Preorder Traversal • Root, left, right • Example: • F, B, A, D, H, K F B H KDA
- 11. Preorder Traversal • Example 2, p-714
- 12. Postorder Traversal • Let T be an ordered rooted tree with root r and T1, T2, …Tn are the subtrees at r from left to right. The preorder traversal begins by traversing T1 in postorder, then T2 in postorder, then T3 in postorder, . . . , and finally Tn in postorder and ends by visiting r.
- 13. Postorder Traversal • Left, right, root • Example: • A, D, B, K, H, F F B H KDA
- 14. Postorder Traversal • Example 4, p-714 02/09/15
- 15. A Shortcut for Traversing an Ordered Rooted Tree in Preorder, Inorder, and Postorder. • See Q. 7 to 15, p-723