Uses for Trees About Trees Binary Trees. Trees. Seth Long. January 31, 2010
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1 Uses for About Binary January 31, 2010
2 Uses for About Binary Uses for Uses for About Basic Idea Implementing Binary Example: Expression Binary Search
3 Uses for Uses for About Binary Uses for Storage Binary Search B-trees (reiserfs) Search Game playing Troubleshooting problems Machine learning: Decision trees Expression evaluation Compilation of programs
4 Basic Idea Uses for About Binary Basic Idea Implementing A tree is a set of nodes where each node has exactly one parent (except for the root) Any sub-tree is a tree An edge is a connection between two nodes In a tree, all edges are directed A node is a child of it s parent Nodes with no children are leaves Nodes with the same parent are siblings A path from node n 1 to n k is a sequency of nodes n 1, n 2, n 3...n k such that n i is the parent of n i+1 for 1 i < k The length of a path is the number of edges on the path (k 1) There is exactly one path from the root to each node n i...n k are descendants of n i and ancestors of n k ni is not a proper descendant, n k is not a proper ancestor
5 Basic Idea Continued Uses for About Binary Basic Idea Implementing The depth of a node is the length of the unique path from the root to that node The root node has a depth of 0 The depth of a tree is the depth of it s deepest leaf The height of a node is the length of the longest path from that node to a leaf All leaves have a height of 0 The height of a tree is the height of it s root node The height of a tree equals it s depth
6 Implementing Uses for About Binary Basic Idea Implementing Idea 1: Vector of children Use an STL vector to hold children O(1) access to children by number Idea 2: List of children Can insert children quickly Can traverse children efficiently, but not find them by number Idea 3: STL map of children Yes, an STL map Quick access to children by name Binary tree: Just use pointers Generally, a node is some data and a set of references to children
7 Uses for About Binary Example: Expression Binary Search Expression Each node has at most two children In this example, no trinary operators! This is often the case Evaluate from bottom up in a recursive manner Example: Expression tree for (a + b c) + ((d e + f ) g)
8 Uses for About Binary Example: Expression Binary Search Binary Search Search in O(h) time Where h is the height of the tree and is proportional to log n in a balanced tree Worst case for badly unbalanced tree is O(n) You already know how to search in binary trees - I ll spare you the examples Traversal: In-order traversal: left, root, right Pre-order traversal (Same as depth-first search): root, left, right Post-order traversal: left, right, root Breadth-first search: Down from the top
9 Uses for About Binary Example: Expression Binary Search Inserting into BST Complications: Need to find a node with less than two children General idea: Either the node before or after the one to be inserted will have a free place This is always true. Why? Every other node is a leaf in a full tree. Complexity?
10 Uses for About Binary Example: Expression Binary Search Removing nodes If 0 or 1 child, remove it. Example (remove 4): If 2 children, replace with successor, then remove it. Example (remove 2):
11 Uses for About Binary Example: Expression Binary Search Randomally generated 500-node tree (insert only)
12 Uses for About Binary Example: Expression Binary Search After many insert/remove pairs
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