09-Trees - 3/1/2011 Tree Overview CS/ENGRD 2110...

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3/1/2011 1 CS/ENGRD 2110 Object-Oriented Programming and Data Structures Spring 2010 Thorsten Joachims Lecture 9: Trees Tree Overview Tree : recursive data structure (similar to list) Each cell may have zero or more successors (children) Each cell has exactly one predecessor (parent) except the root , which has none Cells without children are called leaves All cells are reachable from root Binary tree : tree in which each cell can have at most two children: a left child and a right child 2 5 4 7 8 9 2 General tree 5 4 7 8 2 Binary tree 5 4 7 8 Not a tree 5 6 8 List-like tree Tree Terminology M is the root of this tree G is the root of the left subtree of M B, H, J, N, and S are leaves N is the left child of P; S is the right child P is the parent of N M and G are ancestors of D P, N, and S are descendants of W Node J is at depth 2 (i.e., depth = length of path from root = number of edges) Node W is at height 2 (i.e., height = length of longest path to a leaf) A collection of several trees is called a ...? 3 M G W P J D N H B S Class for Binary Tree Cells class TreeCell<T> { private T datum; private TreeCell<T> left, right; public TreeCell(T x) { datum = x; left = null; right = null; } public TreeCell(T x, TreeCell<T> lft, TreeCell<T> rgt) { datum = x; left = lft; right = rgt; } more methods: getDatum, setDatum, getLeft, setLeft, getRight, setRight } 4 ... new TreeCell<String>("hello") . .. Class for General Trees class GTreeCell { private Object datum; private GTreeCell left; private GTreeCell sibling; appropriate getter and setter methods } 5 5 4 7 8 9 2 7 8 3 1 5 4 7 8 9 2 7 8 3 1 General tree Tree represented using GTreeCell Parent node points directly only to its leftmost child Leftmost child has pointer to next sibling, which points to next sibling, etc. Applications of Trees Most languages (natural and computer) have a recursive, hierarchical structure This structure is implicit in ordinary textual representation Recursive structure can be made explicit by representing sentences in the language as trees: Abstract Syntax Trees (ASTs) ASTs are easier to optimize, generate code from, etc. than textual representation A parser converts textual representations to AST 6
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3/1/2011 2 Example Expression grammar: E → integer E → (E + E) In textual representation Parentheses show hierarchical structure In tree representation Hierarchy is explicit in the structure of the tree
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09-Trees - 3/1/2011 Tree Overview CS/ENGRD 2110...

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