Discrete Structures

Lecture 13: TreesJi Yanyan

United International College

Thanks to Professor Michael Hvidsten

Game Trees

• Definition: A Game Tree models the different outcomes possible in a game.

Vertices: positions in a game

Edges: legal moves from one position to another

Leaves: Final positions of a game

Game Trees

• Example: Tic-Tac-Toe

10.3 Tree Traversal

• Rooted trees are used to store information. We often need to do some operation on the vertices in such a tree. A tree traversal is an algorithm designed to “visit” each node in the tree.

• Traversal Algorithms: – Preorder traversal– Inorder traversal– Postorder traversal

• These types correspond to Prefix/Infix/Postfix notation.

10.3 Tree Traversal

• Definition: Let T be a rooted tree with root r. If T consists only of r, then r is the preorder traversal of T. Otherwise, let T1 , T2 , … , Tn be the subtrees at r from left to right. The preorder traversal of T will begin by visiting r, then T1 (in preorder), then T2 (in preorder), etc, until Tn is traversed in preorder.

10.3 Tree Traversal

10.3 Tree Traversal

• Preorder Traversal = (a,b,e,j,k,n,o,p, f,c,d,g,l,m,h,i)

10.3 Tree Traversal

• Definition: Let T be a rooted tree with root r. If T consists only of r, then r is the inorder traversal of T. Otherwise, let T1 , T2 , … , Tn be the subtrees at r from left to right. The inorder traversal of T will begin by visiting T1 (inorder), then the root r, then T2 (inorder), etc, until Tn is traversed inorder.

10.3 Tree Traversal

10.3 Tree Traversal

• Inorder Traversal = (j,e,n,k,o,p,b,f,a, c,l,g,m,d,h,i)

10.3 Tree Traversal

• Definition: Let T be a rooted tree with root r. If T consists only of r, then r is the postorder traversal of T. Otherwise, let T1 , T2 , … , Tn be the subtrees at r from left to right. The postorder traversal of T will begin by visiting T1 (in postorder), then T2 (in postorder), etc, then Tn (in postorder) and finally the root r.

10.3 Tree Traversal

10.3 Tree Traversal

• Postorder Traversal = (j,n,o,p,k,e,f,b, c,l,m,g,h,i,d,a)

Expression Trees

• The expression tree for an arithmetic expression consists of – Vertices: numbers, +, -, *, /, ↑

• (↑ represents the power function) – Edges: linking parts of an expression– Internal vertices represent operations– Leaves represent the variables or numbers

Expression Trees

• Example: We build the expression tree for ((x+y)↑2)+ ((x-4)/3) from the “bottom” up

Expression Trees

• Infix form of the expression = inorder traversal

• Get: x+y↑2+x-4/3• Add parentheses for groups:• ((x+y)↑2)+((x-4)/3)

Expression Trees

• Prefix form of the expression = preorder traversal

• Get: +↑+ x y 2 / - x 4 3

Expression Trees

• What is expression with prefix form + - * 2 3 5 /↑2 3 4 ?• Answer: ((2*3)-5)+((2↑3)/4)

Expression Trees

• Postfix form of the expression = postorder traversal

• Get: x y + 2 ↑ x 4 – 3 / +

10.4 Spanning Trees

• Recall: A tree is an undirected connected graph without cycles

• Definition: A spanning tree of a connected undirected graph G is a subgraph of G that contains all of G’s vertices and enough of its edges to form a tree

• To obtain a spanning tree from a connected undirected graph with cycles– Remove edges until there are no cycles

10.4 Spanning Trees

10.4 Spanning Trees

• There are two algorithms for constructing spanning trees:

• Depth-First Search (Back-tracking)• Breadth-First Search

Depth-First Search

• Depth-First Search (DFS) proceeds along a path from a vertex v as deeply into the graph as possible before backing up (back-tracking)

• To create a depth-first search (DFS) spanning tree– Traverse the graph using a depth-first search and

mark the edges that you follow– After the traversal is complete, the graph’s vertices

and marked edges form the spanning tree

Depth-First Search

Start at f:

Breadth-First Search

• Breadth-First Search (BFS) visits every vertex adjacent to a vertex v that it can before visiting any other vertex

• To create a breath-first search (BFS) spanning tree– Traverse the graph using a bread-first search and

mark the edges that you follow– When the traversal is complete, the graph’s

vertices and marked edges form the spanning tree

Breadth-First Search

Start at e

Minimum Spanning Trees

• Definition: A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible sum of weights of edges. – There may be several minimum spanning trees for

a particular graph

Prim’s Algorithm

• Prim’s Algorithm finds a minimal spanning tree that begins at any vertex– Find the least-cost edge (v, u) from a visited vertex

v to some unvisited vertex u– Mark u as visited– Add the vertex u and the edge (v, u) to the

minimum spanning tree– Repeat the above steps until there are no more

unvisited vertices

Prim’s Algorithm

• Prim’s Algorithm finds a minimal spanning tree that begins at any vertex– Find the least-cost edge (v, u) from a visited vertex

v to some unvisited vertex u– Mark u as visited– Add the vertex u and the edge (v, u) to the

minimum spanning tree– Repeat the above steps until there are no more

unvisited vertices

Prim’s Algorithm

Kruskal’s Algorithm

• Kruskal’s Algorithm finds a minimal spanning tree that begins at any vertex– Choose a least-cost edge (v, u) as the root– Successively add edges with minimum weight that

do not from a circuit with already chosen edges. – Stop after n-1 edges are chosen.

Kruskal’s Algorithm

Ass 14

• P723 17(b)(c)(d)• Use breadth-first and depth-first search to produce a

spanning tree for the simple graphs in Exercises 13 of page 735. Choose e as the root of each spanning tree.( Using the alphabet order)

Bonus: P723 23(a)(c)