Lecture 17 Trees

1
Lecture 17 Trees

• July 21st , 2003

2
Tree--Definition

• A tree is an acyclic, connected graph with one
  node designated as the root of the tree.
• Special case
• nonrooted tree (free tree)
• Forest acyclic graph (not necessarily
  connected)
• Parent vs child(ren).

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Definition (cont.)

• Depth
• Leaf
• Node
• Binary tree
• Full binary tree
• Complete binary tree (almost full).

4
Tree -- Applications

• In data structure
• Binary search tree
• 2-3 tree
• In file management
• Files in computer are organized in a tree-like
  structure.
• In biology
• Family tree

5
Creating EXPRESSION Tree

• Ex. 23 mentions the idea and shows an
• Expression Tree for
• (2 x) (y 3)
• --- MINUS is the ROOT node PLUS MULT
  first-level below the root
• and 2, x, y, 3 are the LEAVES
• Prax 19 p. 374 asks for an ExpTree for a
  simpler case (2 3) 5
• --- When youve seen one Expressn Tree youve
  seem em all ?

6
Binary tree representation

• Left child - right child rep --- TABLE (See lotsa
  0,0 pairs!!)
• Pointer representation --- Linked List (3-Slot
  Elements)
• Ex 24 p. 374-375 illustrates both these REPs
  for a 3-level binary tree with one missing leaf
  !! 1 ? 2, 3 2 ? 4, 5 3 ? 6
• Prax. 20 p. 375 illustrates both these REPs
  for a 4-level tree which opens only to one side
  (from the top node)
• COMMENT on Prax 20 By writeg out the
  solutions in the white space of the page where
  Prax 20 is found, I thought it would be better to
  FIRST do the POINTER rep and then follow it up
  with Left-Child/Right-Child solution I
  report-you decide The pointer rep. kinda
  matches up with the drawing and helps identify
  0 (or ) locations for the LChild-RChild Rep

7
Tree traversal algorithms

• Pre-order --- Root-Left-Right .. RoLtRt e-o
  o-e
• In-order --- Left-Root-Right.. LtRoRt
  i-o e-o
• Post-order --- Left-Right-Root.. LtRtRo o-o
  e-i
• (Pre- and Post- are opposites, i.e. Ro-LR vs
  LR-Ro and In puts Ro in-side?)
• A basic idea is a recursive view of a tree
• Subtrees are labelled as T1 , T2, ,Tt

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A traversal algorithm

• ALGORITHM PREORDER
• Preorder (tree T)
• write (r)
• for i1 to t do
• Preorder(Ti)
• end for
• end Preorder
• Postorder merely shifts the write the for
  loop (and changes names)
• Inorder mimics Preorder mit an extra call on
  Inorder(Ti) at the start also changeg the for
  loop start to 2
• See non binary tree remarks on p. 378

9
Traversal--Examples

• Ex 25-27 p377 Ar puts ans. right above the
  Tree for PRE-ORD (Root-LR)
• a
• b
  c
• d e f
  g
• h
  i
• and then just lists (on the page below) the
  solutions for INORD (L-Root-R, first, and POSTORD
  (LR-Root) last
• Prax 21- p. 378 does about the same thing (all 3
  orders)
• Can we rep. algebra expressions as binary trees?
  Choices are given as
• Get Infix notation from Inorder, uncoverg (2
  x) 4 --- PARENS put in (How?)
• Git Prefix notation (aka Polish notation) from
  Preorder .. uncoverg 2 x 4
• Get Postfix notation (aka Reverse Polish)..
  Uncoverg 2 x 4
• ? See a Pre- Post- convert to Infix (a
  hint on how parens appear?)
• b Pre- Post- DO NOT NEED PARENS
  at all (SEMANTIC-wise) and
• c They evaluate sequentially
  mit NO LOOK-AHEADs for paren-exprs

10
Some properties of trees

• A tree with n nodes has n-1 arcs
• Any tree with n nodes, the total number of arc
  ends is 2n-2.
• A binary tree has at most 2d nodes at depth d.
• The number of leaves in any binary tree is 1 more
  than the number of nodes with two children.

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Exercise

• Exercises 5.2
• 3, 10,23, 28, 38, 40, 41