7.1 Introduction. A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w

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1 Chapter 7 Trees

2 7.1 Introduction A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w

3 Tree Terminology Parent Ancestor Child Descendant Siblings Terminal vertices Internal vertices Subtrees

4 Rooted tree A rooted tree is a tree where one of its vertices is designated the root

5 Rooted Trees Example: Family Tree Rooted Tree: a directed graph T such that 1. If ignore direction of edges, graph is a tree 2. There exists unique vertex R with in-degree 0, in-degree of all other V is 1 (R is called the Root) Usually draw with Root at top & other edges going down

6 Theorem 1 Let (T,v 0 ) be a rooted tree. Then: 1. No Directed Cycles 2. V 0 is the only root of T. 3. Each vertex in T, other than the root V 0 has indegree one, and the root has in-degree of zero.

7 Tree Terminology (3) Subtree of a node: A tree whose root is a child of that node Level of a node: A measure of its distance from the root: Level of the root = 1 Level of other nodes = 1 + level of parent

8 Section 7.2 Labeled Trees

9 Arithmetic expressions Standard: infix form (A+B) * C D/ E Fully parenthesized form (inorder & parenthesis): (((A + B) * C) (D / E)) Postfix form (reverse Polish notation): A B + C * D E / - Prefix form (Polish notation): - * + A B C / D E

10 Binary Trees & Traversals A rooted tree in which each vertex has at most 2 children, denoted left child & right child Left subtree of V: subtree rooted by left child of V Right subtree

11 Binary trees A binary tree is a tree where each vertex has zero, one or two children

12 Full binary tree A full binary tree is a binary tree in which each vertex has two or no children.

13 Binary Search Tree (BST) Consider each vertex has a value. BST: For every vertex, all vertices in left subtree have greater value that root and all vertices in right subtree have values less than root.

14 Binary search trees Data are associated to each vertex Order data alphabetically, so that for each vertex v, data to the left of v are less than data in v and data to the right of v are greater than data in v Example: "Computers are an important technological tool"

15 Huffman codes On the left tree the word rate is encoded On the right tree, the same word rate is encoded

16 Examples of Binary Trees

17 Examples of Binary Trees Code for b = Code for w = Code for s = 0011 Code for e = 010

18 Example 3

19 Example 4

20 Section 7.3 Tree Searching

21 Traversals of Binary Trees Often want iterate over and process nodes of a tree Can walk the tree and visit the nodes in order This process is called tree traversal Three kinds of binary tree traversal: Preorder Inorder Postorder According to order of subtree root w.r.t. its children

22 Binary Tree Traversals (2) Preorder: Visit root, traverse left, traverse right Inorder: Traverse left, visit root, traverse right Postorder: Traverse left, traverse right, visit root

23 Visualizing Tree Traversals Can visualize traversal by imagining a mouse that walks along outside the tree If mouse keeps the tree on its left, it traces a route called the Euler tour: Preorder: record node first time mouse is there Inorder: record after mouse traverses left subtree Postorder: record node last time mouse is there

24 Visualizing Tree Traversals (2) Preorder: a, b, d, g, e, h, c, f, i, j

25 Visualizing Tree Traversals (3) Inorder: d, g, b, h, e, a, i, f, j, c

26 Visualizing Tree Traversals (4) Postorder: g, d, h, e, b, i, j, f, c, a

27 Traversals of Binary Search Trees Inorder traversal of a binary search tree à Nodes are visited in order of increasing data value Inorder traversal visits in this order: canine, cat, dog, wolf

28 Traversals of Expression Trees Inorder traversal can insert parentheses where they belong for infix form Postorder traversal results in postfix form Prefix traversal results in prefix form Infix form Postfix form: x y + a b + c / * Prefix form: * + x y / + a b c

29 Section 7.5 Minimal spanning tree

30 Spanning trees Given a graph G, a tree T is a spanning tree of G if: T is a subgraph of G and T contains all the vertices of G

31 Spanning tree search Breadth-first search method Depth-first search method (backtracking)

32 Minimal spanning trees Given a weighted graph G, a minimum spanning tree is a spanning tree of G that has minimum weight

33 1. Prim s algorithm Step 0: Pick any vertex as a starting vertex (call it a). T = {a}. Step 1: Find the edge with smallest weight incident to a. Add it to T Also include in T the next vertex and call it b. Step 2: Find the edge of smallest weight incident to either a or b. Include in T that edge and the next incident vertex. Call that vertex c. Step 3: Repeat Step 2, choosing the edge of smallest weight that does not form a cycle until all vertices are in T. The resulting subgraph T is a minimum spanning tree.

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected.

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected. Chapter 4 Trees 4-1 Trees and Spanning Trees Trees, T: A simple, cycle-free, loop-free graph satisfies: If v and w are vertices in T, there is a unique simple path from v to w. Eg. Trees. Spanning trees: