Binary Search Tree--Data Structure

Hash table

RangeSearch: impossible

NearestNeighbors: impossible

Insert:   O(1)

Delete:  O(1)

Array:

RangeSearch:        O(n)

NearestNeighbors: O(n)

Insert:                     O(1)

Delete:                    O(1)

Sorted Array:

RangeSearch:        O(logn)

NearestNeighbors: O(logn)

Insert:                     O(n)

Delete:                    O(n)

Linked List:

RangeSearch:        O(n)

NearestNeighbors: O(n)

Insert:                     O(1)

Delete:                    O(1)

Search Tree Property

X’s key is larger than the key of any descendent of its left child, and smaller than the key of any descendant of its right child.

Operations:

Find(k,R)

if R.Key=k:

  return R

else if R.Key>k:

  return Find(k,R.Left)

else if R.Key<k:

  return Find(k,R.Right)

Find (modified)

else if R.Key>k:

  if R.Left!=null:

    return Find(k,R.Left)

return R

Next(N)

if N.Right!=null:
  return LeftDescendant(N.Right)

else:
  return RightAncestor(N)

LeftDescendant(N)

if N.Left=null

  return N

else:
  return LeftDescendant(N.Left)

RightAncestor(N)

if N.Key<N.Parent.Key

  return N.Parent

else:
  return RightAncestor(N.Parent)

RangeSearch(x,y,R)

L←∅
N ←Find(x,R)

while N.Key≤y

  if N.Key≥x:
    L ←L.Append(N)

  N ←Next(N)

return L

Insert(k,R)

P ←Find(k,R)
Add new node with keykas child  of P

Delete(N)

if N.Right=null:
  Remove N, promoteN.Left

else:
  X ←Next(N)

\\   X.Left=null
  Replace N byX, promoteX.Right

RotateRight(X)

P ←X.Parent
Y ←X.Left
B ←Y.Right
Y .Parent←P

P.AppropriateChild←Y

X.Parent←Y, Y.Right←X

B.Parent←X, X.Left←B