Trees and Tree Traversal

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Liliana Norris
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1 Trees and Tree Traversal Material adapted courtesy of Prof. Dave Matuszek at UPENN Definition of a tree A tree is a node with a value and zero or more children Depending on the needs of the program, the children may or may not be ordered A B C D E F G H I J K L M N A tree has a root, internal nodes, and leaves Each node contains an element and has branches leading to other nodes (its children) Each node (other than the root) has a parent Each node has a depth (distance from the root) 2 1

2 More definitions An empty tree has no nodes The descendents of a node are its children and the descendents of its children The ancestors of a node are its parent (if any) and the ancestors of its parent The subtree rooted at a node consists of the given node and all its descendents An ordered tree is one in which the order of the children is important; an unordered tree is one in which the children of a node can be thought of as a set The branching factor of a node is the number of children it has The branching factor of a tree is the average branching factor of its nodes 3 File systems File systems are almost always implemented as a tree structure The nodes in the tree are of (at least) two types: folders (or directories), and plain files A folder typically has children subfolders and plain files A plain file is typically a leaf 4 2

3 Family trees It turns out that a tree is not a good way to represent a family tree Every child has two parents, a mother and a father Parents frequently remarry An upside down binary tree almost works Since it is a biological fact (so far) that every child has exactly two parents, we can use left child = father and right child = mother The terminology gets a bit confusing If you could go back far enough, it becomes a mathematical certainty that the mother and father have some ancestors in common 5 Part of a genealogy Chester Elaine Eugene Pauline David Paula Winfred Carol Steven Danielle Isaac 6 3

4 Game trees Trees are used heavily in implementing games, particularly board games A node represents a position on the board The children of a node represent all the possible moves from that position More precisely, the branches from a node represent the possible moves; the children represent the new positions Planning ahead (in a game) means choosing a path through the tree However You can t have a cycle in a tree If you can return to a previous position in a game, you have a cycle Graphs can have cycles 7 Tree searches B A C A tree search starts at the root and explores nodes from there, looking for a goal node (a node that satisfies certain conditions, depending on the problem) D E F H I L M N O P J G K Q For some problems, any goal node is acceptable (N or J); for other problems, you want a minimum-depth goal node, that is, a goal node nearest the root (only J) Goal nodes 8 4

5 Depth-first searching A B D E F H I L M N O P C J G K Q A depth-first search (DFS) explores a path all the way to a leaf before backtracking and exploring another path For example, after searching A, then B, then D, the search backtracks and tries another path from B Node are explored in the order A B D E H L M N I O P C F G J K Q N will be found before J 9 How to do depth-first searching Put the root node on a stack; while (stack is not empty) { remove a node from the stack; put all children of node onto the stack; At each step, the stack contains some nodes from each of a number of levels The size of stack that is required depends on the branching factor b While searching level n, the stack contains approximately b*n nodes 10 5

6 Recursive depth-first search Recursion is always implemented with a stack. Anything you can do with recursion, you can do with stacks, and vice-versa search(node): print node and if node is a goal, return success; for each child c of node { print c and if search(c) is successful, return success; 11 Recursive depth-first search search(node): print node and if node is a goal, return success; for each child c of node { print c and if search(c) is successful, return success; The (implicit) stack contains only the nodes on a path from the root to a goal When a solution is found, the path is on the (implicit) stack, and can be extracted as the recursion unwinds You can do this by (for example) substituting pushing values onto a stack for the above print statements 12 6

7 Breadth-first searching D B E A F C G A breadth-first search (BFS) explores nodes nearest the root before exploring nodes further away For example, after searching A, then B, then C, the search proceeds with D, E, F, G H I J K Node are explored in the order A B C D E F G H I J K L M N O P Q J will be found before N L M N O P Q 13 How to do breadth-first searching Put the root node on a queue; while (queue is not empty) { remove a node from the queue; put all children of node onto the queue; 14 7

8 Comparison of algorithms Depth-first searching: Put the root node on a stack; while (stack is not empty) { remove a node from the stack; put all children of node onto the stack; Breadth-first searching: Put the root node on a queue; while (queue is not empty) { remove a node from the queue; put all children of node onto the queue; 15 How to do breadth-first searching Put the root node on a queue; while (queue is not empty) { remove a node from the queue; put all children of node onto the queue; Just before starting to explore level n, the queue holds all the nodes at level n-1 In a typical tree, the number of nodes at each level increases exponentially with the depth Memory requirements may be infeasible When this method succeeds, it doesn t give the path There is no recursive breadth-first search equivalent to recursive depth-first search 16 8

9 Depth- vs. breadth-first searching When a breadth-first search succeeds, it finds a minimum-depth (nearest the root) goal node When a depth-first search succeeds, the found goal node is not necessarily minimum depth For a large tree, breadth-first search memory requirements may be excessive For a large tree, a depth-first search may take a very long time to find even a very nearby goal node There is a way to combine the two, but it is beyond the scope of this class. Channel your inner Googler to look up Depth-limited searching and iterative deepening. 17 Thank you! 18 9

Definition of a tree. A tree is like a binary tree, except that a node may have any number of children

Definition of a tree. A tree is like a binary tree, except that a node may have any number of children Trees Definition of a tree A tree is like a binary tree, except that a node may have any number of children Depending on the needs of the program, the children may or may not be ordered Like a binary tree,