Trees, Part 1

1. Trees, Part 1 CSCI 2720 University of Georgia Spring 2007

2. Tree Properties and Definitions • Composed of nodes and edges • Node • Any distinguishable object • Edge • An ordered pair <u,v> of nodes • Illustrated by an arrow with tail at u and head at v • u = tail of <u,v> • v = head of <u,v> • Root • A distinguished node, depicted at the top of the tree

3. Recursive Definition of a Tree • A single node, with no edges, is a tree. The root of the tree is its unique node. • Let T1, …, Tk (k >=1) be trees with no nodes in common, and let r1, …,rk be the roots of those trees, respectively. Let r be a new node. Then there is a tree T consisting of the nodes and edges of T1, …, Tk , the new node r and the edges <r,r1>,…,<r,rk>. The root of T is r and T1, …,Tk are called the subtrees of T.

4. Tree definitions and terminology • r is called the parent of r1, …, rk • r1, …, rk are the children of r • r1, …, rk are the siblings of one another • Node v is a descendant of u if • u = v or • v is a descendant of a child of u • A path exists from u to v.

5. Recursive definition Before: The trees T1, …, Tk, with roots r1, …, rk After: new tree T rooted at r with subtreesT1, …, Tk,

6. What do we mean by a path from u to v? • You can get from u to v by following a sequences of edges down the tree, starting with an edge whose tail is u, ending with an edge whose head is v, and with the head of each edge but the last being the tail of the next.

7. Tree terminology • Every node is a descendant of itself. • If v is a descendant of u, then u is an ancestor of v. • Properdescendants: • The descendants of a node, other than the node itself. • Proper ancestors • The ancestors of a node, other than the node itself.

8. Tree terminology & properties • Leaf • a node with no children • Num_nodes = Num_edges + 1 • Every tree has exactly one more node than edge • Every edge, except the root, is the head of exactly one of the edges

9. Tree terminology & properties • Height of a node in a tree • Length of the longest path from that node to a leaf • Depth of a node in a tree • Length of the path from the root of the tree to the node

10. Tree terminology root root leaf leaf leaf leaf leaf leaf

11. Tree terminology u v A path from u to v. Node u is the ancestor of node v.

12. Tree terminology • w has depth 2 • u has height 3 • tree has height 4 u w

13. Special kinds of trees • Ordered v. unordered • Binary tree • Empty v non-empty • Full • Perfect • Complete

14. Ordered trees • Have a linear order on the children of each node • That is, can clearly identify a 1st, 2nd, …, kth child • An unordered tree doesn’t have this property

15. Binary tree • An ordered tree with at most two children for each node • If node has two child, the 1st is called the left child, the 2nd is called the right child • If only one child, it is either the right child or the left child Two trees are not equivalent …One has only a left child; the other has only a right child

16. Empty binary tree • Convenient to define an empty binary tree, written , for use in this definition: • A binary tree is either  or is a node with left and right subtrees, each of which is a binary tree.

17. Full binary tree • Has no nodes with only one child • Each node either a leaf or it has 2 children • # leaves = #non-leaves + 1

18. Perfect binary tree • A full binary tree in which all leaves have the same depth • Perfect binary tree of height h has: • 2h+1 -1 nodes • 2h leaves • 2h -1 non-leaves • Interesting because it is “fully packed”, the shallowest tree that can contain that many nodes. Often, we’ll be searching from the root. Such a shallow tree gives us minimal number of accesses to nodes in the tree.

19. Perfect binary trees

20. Complete binary tree • Perfect binary tree requires that number of nodes be one less than a power of 2. • Complete binary tree is a close approximation, with similar good properties for processing of data stored in such a tree. • Complete binary tree of height h: • Perfect binary tree of height h-1 • Add nodes from left at bottom level

21. More formally, … • A complete binary tree of height 0 is any tree consisting of a single node. • A complete binary tree of height 1 is a tree of height 1 with either two children or a left child only. • For height h >= 2, a complete binary tree of height h is a root with two subtrees satisfying one of these two conditions: • Either the left subtree is perfect of height h-1 and the right is complete of height h-1 OR • The left is complete of height h-1 and the right is perfect of height h-2.

22. Complete binary trees

23. Tree terminology • Forest • A finite set of trees • In the case of ordered trees, the trees in the forest must have a distinguishable order as well.

24. Tree Operations • Parent(v) • Children(v) • FirstChild(v) • RightSibling(v) • LeftSibling(v) • LeftChild(v) • RightChild(v) • isLeaf(v) • Depth(v) • Height(v)

25. Parent(v) • Return the parent of node v, or  if v is the root.

26. Children(v) • Return the set of children of node v (the empty set, if v is a leaf).

27. FirstChild(v) • Return the first child of node v, or  if v is a leaf.

28. RightSibling(v) • Return the right sibling of v, or  if v is the root or the rightmost child of its parent.

29. LeftSibling(v) • Return the left sibling of v, or  if v is the root or the leftmost child of its parent

30. LeftChild(v) • Return the left child of node v; • Return  if v has no left or right child.

31. RightChild(v) • Return the right child of node v; • return  if v has no right child

32. isLeaf(v) • Return true if node v is a leaf, false if v has a child

33. Depth(v) • Return the depth of node v in the tree.

34. Height(v) • Return the height of node v in the tree.

35. Arithmetic Expression Tree • Binary tree associated with an arithmetic expression • internal nodes: operators • external nodes: operands • Example: arithmetic expression tree for the infix expression: • (2 × (a − 1) + (3 × b)) • Note: infix expressions require parentheses • ((2 x a) – (1+3) * b) is a different expression tree

36. Evaluating Expression Trees function Evaluate(ptr P): integer /* return value of expression represented by tree with root P */ if isLeaf(P) return Label(P) else xL <- Evaluate(LeftChild(P)) xR <- Evaluate(RightChild(P)) op <- Label(P) return ApplyOp(op, xL, xR)

37. Example - Evaluate (A) Evaluate(B) Evaluate(D) => 2 Evaluate(E) Evaluate(H)=> a Evaluate(I) => 1 ApplyOp(-,a,1) => a-1 ApplyOp(x,2,(a-1)) => (2 x (a-1)) Evaluate(C) Evaluate(F) => 3 Evaluate(G) => b ApplyOp(x,3,b) => 3 x b ApplyOp(+,(2 x (a-1)),(3 x b)) => (2 x (a-1)) + (3 x b) A B C F G E D H I

38. Traversals • Traversal = any well-defined ordering of the visits to the nodes in a tree • PostOrder traversal • Node is considered after its children have been considered • PreOrder traversal • Node is considered before its children have been considered • InOrder traversal (for binary trees) • Consider left child, consider node, consider right child

39. PostOrder Traversal procedurePostOrder(ptr P): foreach childQ of P, in order, do PostOrder(Q) Visit(P) PostOrder traversal gives: 2, a, 1, -, x, 3, b, x, + Nice feature: unambiguous … doesn’t need parentheses -- often used with scientific calculators -- simple stack-based evaluation

40. Evaluating PostOrder Expressions procedurePostOrderEvaluate(e1, …, en): forifrom 1 to ndo ifei is a number, then push it on the stack else • Pop the top two numbers from the stack • Apply the operator ei to them, with the right operand being the first one popped • Push the result on the stack

41. PreOrder Traversal procedurePreOrder(ptr P): Visit(P) foreach childQ of P, in order, do PreOrder(Q) PreOrder Traversal: +, x, 2, -, a, 1, x, 3, b Nice feature: unambigous, don’t need parentheses -- Outline order

42. InOrder Traversal procedureInOrder(ptr P): // P is a pointer to the root of a binary tree if P =  then return else InOrder(LeftChild(P)) Visit(P) InOrder(RightChild(P)) InOrder Traversal: 2 x a – 1 + 3 x b

43. Binary Search Trees • Use binary trees to store data that is linearly ordered • A linear order is a relation < such that for any x, y, and z • If x and y are different then either x<y or y<x • If x<y and y < z then x< z • An inorder traversal of a binary search tree yields the labels on the nodes, in linear order.

44. Binary Search Trees, examples 50 60 30 70 20 40 Inorder traversal yields: 20, 30, 40, 50, 60, 70

45. Level-order traversal a.k.a. breadth-first traversal A C B F D E breadth-first or level-order traversal yields: A, B, C, D, E, F

46. Tree Implementations • General Binary Tree • General Ordered Trees • Complete Binary Trees

47. Binary Trees:“Natural Representation” • LC : contains pointer to left child • RC: contains pointer to right child • Info: contains info or pointer to info • LeftChild(p), RightChild(p): (1) • Parent(p): not (1) unless we also add a parent pointer to the node LC info RC • Parent(p): not (1) unless we also add a parent pointer to the node • Can use xor trick to store parent-xor-left, • Parent-xor-right, and use only two fields

48. Ordered Tree representation • Binary tree: array of 2 ptrs to 2 children • Ternary tree: array of 3 ptrs to 3 children • K-ary trees: k pointers to k children • What if there is no bound on the number of children? • Use binary tree representation of ordered tree: First Info Right Child Sibling

49. General ordered tree Binary tree rep Example A A B B C D C E D F E F G H I G H I

50. Complete Binary Tree: Implicit representation Binary Search Trees, example 50 70 30 20 40 60 0 1 2 3 4 5 [50 | 30 | 70 | 20 | 40| 60]