Copyright © Curt Hill Other Trees Applications of the Tree Structure

Copyright © Curt Hill 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

Copyright © Curt Hill Expression trees 2+3*4 + * *2 34 3*4+2

Copyright © 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

Copyright © 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 * 4 This gives standard algebraic notation

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

Copyright © Curt Hill Postorder Subtrees first and then node Visit lower (left) subtree Visit upper (right) subtree Process node Expression: * + Reverse Polish

Parse Trees Expression trees are a small instance of parse trees A presentation on parse trees also existspresentation on parse trees Copyright © 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

Copyright © Curt Hill Unbalanced Tree

Copyright © 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

Copyright © Curt Hill Perfectly Balanced Tree

Copyright © Curt Hill Average Search Length 36 – 1 4 – 2 24 – 2 2 – – 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

Copyright © Curt Hill AVL Balanced Tree

Copyright © Curt Hill Average Search Length 36 – 1 6 – 2 19 – 2 2 – – 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

Copyright © 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

Copyright © 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

Copyright © Curt Hill Average Path Length

Copyright © 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

Copyright © 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

Copyright © 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

Copyright © 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

Copyright © 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

Copyright © 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

Copyright © Curt Hill Example Three element List nil (Three element list)

Copyright © Curt Hill Second Example Atom sub last nil (Atom (sub list) last) list nil

Copyright © 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