Trees

1
Trees
root
internal node
leaf
2
Trees depth
depth 0
depth 1
depth 3
depth 4
depth(root) 0 else depth(n) 1
depth(n.parent())
3
Trees height
height 4
height 0
height 1
height 1
height 0
height 0
height(leaf) 0. else height(n) 1 max(height
of children)
4
Trees ancestry
Green nodes are descendants of red node red node
is ancestor of green nodes. Sometimes we say a
node is an ancestor of itself (e.g. LCA problem
on HW2), but generally not.
5
Trees full
Every node has 0 or two children
6
Trees complete
All levels except possibly the last, are full
last level is left-full (i.e., all nodes as far
left as possible.
7
Animal

• C Is your animal smaller than a breadbox?
• U n
• C Is your animal a cat?
• U n
• C What is your animal?
• U albatross
• C Please enter a question that would be true for
  an albatross, but false for a cat
• U Does your animal have wings?

8
Animal (round 2)

• C Is your animal smaller than a breadbox?
• U n
• C Does your animal have wings?
• U y
• C Is your animal an albatross?
• U n
• C What is your animal?
• U penguin
• C Please enter a question that would be true for
  a penguin, but false for a albatross
• U is your animal earthbound?

9
Initial state
smaller than breadbox?
cat?
10
After round 1
smaller than breadbox?
wings?
cat?
albatross?
11
Another use representing expressions
((1 4) (2 - 5)) 3

3


-
1
4
2
5
12

• class EmptyBinTreeException(Exception)
• class MissingChildException(Exception)
• class Node
• parent(self) // Return parent Node (not
  position)
• left(self) // Return left child Node (not
  position)
• right(self)
• addRight(self, value) //Add right child to this
  node
• // Error if theres one
  already
• addLeft(self, value)
• hasLeft(self) // Return whether the node has a
  left child.
• hasRight(self)
• element(self) // Return the element stored at
  this node
• depth(self) // Return the depth of this node
  in the tree

13

• class Position
• def __init__(self, node)
• self._node node
• def __str__(self)
• return repr(self._node.element())
• def element(self) // PUBLIC part of Position
• return self._node.element()

14

• class BinTree
• root(self)
• parent(self, p) // Return position of parent of
  this position
• children(self, p)// Return posns of children of
  this posn
• isEmpty(self) // Is the tree empty?
• size(self) // Return the size of the tree
• height(self) // Return the height of the tree
• ...

15

• isInternal(self, p) // Return whether posn p
  is internal
• isExternal(self, p)
• isRoot(self, p)
• replace(self, p, e) // replace element at
  position p with e
• left(self, p) // Get posn of left child of
  node at posn p
• // error if no left child
  exists
• right(self, p)
• hasLeft(self, p) // Return whether node at
  posn p has left child.
• hasRight(self, p)
• addRoot(self, e) // Add root to tree Error
  if theres one already
• addLeft(self, p, e) // Add a left child to the
  node at position p // error if theres
  one already
• addRight(self, p, e)

16
Other methods not implemented in HW
iterator(self) return iterator over all
elements stored in tree positions(self)
return iterator over all positions in
tree remove(p) remove position p and replace
it with child, if any. If p has two kids, its
an error attach(p, T1, T2) Attach T1 and T2
as left and right subtrees of leaf at position p.
Error if p is not external