1. Copyright © 2004-2011 Curt Hill Other Trees Applications of the Tree Structure

2. Copyright © 2004-2011 Curt Hill Parse or expression trees An expression tree contains: –Operators as interior nodes –Values as leaves The shape of the expression tree captures the precedence Consider the following expression: 2+3*4

3. Copyright © 2004-2011 Curt Hill Expression trees 2+3*4 + *2 34 + *2 34 3*4+2

4. Copyright © 2004-2011 Curt Hill Notes The plus is always higher in the tree because the precedence of multiplication is higher The expression tree is based upon the grammar rules –Syntax diagrams will be considered later

5. Copyright © 2004-2011 Curt Hill Traversal The names come from the above expression tree There are six (3!) ways to traverse the depending on the order of processing: –The node –The left subtree –The right subtree Inorder (left and right) Preorder Postorder

6. Copyright © 2004-2011 Curt Hill Inorder According to the sorted order of tree Visit lower (left) subtree Process node Visit upper (right) subtree The reverse produces higher to lower Left to right 2 + 3 * 4 This gives standard algebraic notation

7. Copyright © 2004-2011 Curt Hill Preorder Node first then subtrees Process node Visit lower (left) subtree Visit upper (right) subtree Expression + 2 * 3 4 Remember this?

8. Copyright © 2004-2011 Curt Hill Postorder Subtrees first and then node Visit lower (left) subtree Visit upper (right) subtree Process node Expression: 2 3 4 * + Reverse Polish

9. Copyright © 2004-2011 Curt Hill Parse Trees A parse tree is a construction that represents the parse of a sentence Parse trees are often built by compilers and other programs that scan source code A parse tree is one acceptable parse of this source code based on the grammar First some definitions and background

10. Copyright © 2004-2011 Curt Hill A grammar consists of Terminals Keywords Constants Words Symbols or operators Non-terminals Named constructs where the name never actually appears Such as statements and expression Distinguished symbol The starting point Usually represents whole program Productions Rules for going from distinguished symbol to non terminals

11. Copyright © 2004-2011 Curt Hill Example Simplified grammar for expression –Terminals: Constant Ident + - ( ) * / –Productions in EBNF: Expression ::= Term + term Expression ::= term - term Expression ::= term Term ::= Factor [ { * / } factor ] Factor ::= Constant Factor ::= Ident Factor ::= ( Expression )

12. Copyright © 2004-2011 Curt Hill Productions Again The productions are all replacement rules They may also be visually denoted in terms of syntax graphs These are usually as easy to understand but represent same information

13. Copyright © 2004-2011 Curt Hill Syntax Graphs A circle represents a terminal –Reserved word or operator –No further definition A rectangle represents a non-terminal –For statement or expression –Must be defined else where An arrow represents the path between one item and another –The arrows may branch indicating alternatives Recursion is also allowed

14. Copyright © 2004-2011 Curt Hill Simple Expressions expression term + - factor * / constant ident ()expression

15. Copyright © 2004-2011 Curt Hill Parse tree example In a parse tree –Leaves are terminals –Nodes are non-terminals Usually evaluated from bottom to top Consider the parse of: 2 + 5 * ( 3 - 4 )

16. Copyright © 2004-2011 Curt Hill Expression: 2 + 5 * (3 – 4) term- factor 3 term factor 4 expression *factor 5 term + factor 2 expression factor ( (

17. Copyright © 2004-2011 Curt Hill Notes Parse trees are partially built by the parser of the compiler Then used by the code generator Once used the sub-trees are discarded Whole tree never exists

18. Copyright © 2004-2011 Curt Hill Balance and Search Times The time it takes to search a tree is based upon the path length to the desired node Assuming equal distributions then –The average search is the sum of the path lengths divided by the number of tree nodes

19. Copyright © 2004-2011 Curt Hill Unbalanced Tree 12 19 6 2 1536 4 024 30 29

20. Copyright © 2004-2011 Curt Hill Average Search Length 12 – 1 6 – 2 19 – 2 2 – 3 15 – 3 36 – 3 24 – 4 0 – 4 4 – 4 30 – 5 29 – 6 Sum of 37 for 11 nodes gives average search length of 3.3

21. Copyright © 2004-2011 Curt Hill Perfectly Balanced Tree 36 24 4 2 1528 6 01930 12

22. Copyright © 2004-2011 Curt Hill Average Search Length 36 – 1 4 – 2 24 – 2 2 – 3 12 - 3 15 – 3 28 – 3 0 – 4 6 – 4 19 – 4 30 – 4 Sum of 33 fpr 11 nodes gives average search length of 3.0 Balanced does perform better

23. Copyright © 2004-2011 Curt Hill AVL Balanced Tree 36 19 6 2 1528 4 02430 12

24. Copyright © 2004-2011 Curt Hill Average Search Length 36 – 1 6 – 2 19 – 2 2 – 3 12 - 3 15 – 3 28 – 3 0 – 4 4 – 4 28 – 4 30 – 4 Sum of 33 fpr 11 nodes gives average search length of 3.0 AVL balanced has the same performance as perfectly balanced

25. Copyright © 2004-2011 Curt Hill Balanced is Best? The idea of balancing a tree is predicated on equal frequencies of keys –Reasonable assumption if no contrary information –However, if we have frequency information we can do better C++ keywords are not evenly distributed

26. Copyright © 2004-2011 Curt Hill Path Lengths The idea of balance is nice in general but… If we have a reasonable idea of the frequency of entries we can do better than perfectly balanced What we want to do is minimize the average path length With our previous knowledge we could make not assumptions concerning frequency Now we can generate a more precise formula

27. Copyright © 2004-2011 Curt Hill Average Path Length Where –n is the number of words –p is the path length of word i –f is the frequency of word i

28. Copyright © 2004-2011 Curt Hill Optimal Search Trees What we want are high frequency words close to the root and low frequency words at the leaves You might think that the most common word should be the root and the next two words the second and third common It does not work that way since we need to maintain the order as well

29. Copyright © 2004-2011 Curt Hill Example For example the word "the" is the most common word in English text The top n are: –the (20) –and (15) –of (13) –to (12) –you (7) –in (7) –a (6) Because the top two are such extremes it may be better to have “of” as the root

30. Copyright © 2004-2011 Curt Hill LISP Lists LISP is very old –Second only to FORTRAN Usually encountered in Programming Language or Artificial Intelligence classes It has an ubiquitous data structure called a list However it is not a list in the sense that it is purely linear Instead it is a tree, but a tree without a key

31. Copyright © 2004-2011 Curt Hill Variables in LISP A variable may be: –An atom –A list An atom is any word or number A list may be: –Empty –A variable followed by a list

32. Copyright © 2004-2011 Curt Hill Lists A list could be a simple list within parenthesis –(Three element list) It could also have sub-lists –(Atom (A sub list) another (list)) –This is clearly not a linear list such as an STL List LISP programs were also lists –The programs and data had same form

33. Copyright © 2004-2011 Curt Hill Implementation The LISP language was influenced by the machine on which it was developed It had a 36 bit word that was partitioned into two pointers –Contents Address Register (CAR) –Contents Data Register (CDR) An atom used the word for data A list used the pointers and atoms A list always ended in nil, a special pointer

34. Copyright © 2004-2011 Curt Hill Example Three element List nil (Three element list)

35. Copyright © 2004-2011 Curt Hill Second Example Atom sub last nil (Atom (sub list) last) list nil

36. Copyright © 2004-2011 Curt Hill List Processing There were two functions that were continually used in LISP to process a list Car gave the first item of the list –Which could be a list itself Cdr gave the rest of the list A heavy dose of recursion and LISP could do it all