AVL tree--Data Structure

Height

Definition

The heightof a node is the maximum depth of its subtree.

AVL Property

AVL trees maintain the following property:

  For all nodesN, |N.Left.Height− N.Right.Height| ≤ 1

We claim that this ensures balance. 

 

Conclusion

AVL Property

If you can maintain the AVL property, you can perform operations inO(log(n))time.

Insert:

If the balance is destroyed by adding a node, we need to insert nodes to be rebalanced.

AVLInsert(k, R)

Insert(k, R) N ←Find(k,R) Rebalance(N)

 

Rebalance(N)

P ←N.Parent if N.Left.Height> N.Right.Height+1:     RebalanceRight(N) if N.Right.Height> N.Left.Height+1:     RebalanceLeft(N) AdjustHeight(N) if P !=null:     Rebalance(P)

AdjustHeight(N)

N.Height← 1+ max( N .Left.Height,  N.Right.Height)

If subtree too heavy, rotate right.

Delete:

AVLDelete(N)

Delete(N) M ←Parent of node replacing N Rebalance(M)

Summary

AVL trees can implement all of the basic operations inO(log(n))time per operation.

New operations:

Merge:

MergeWithRoot(R1,R2,T )

T.Left← R1 T .Right← R2 R1.Parent← T R2.Parent← T return T

Time O(1). 

Merge(R1,R2)

T ←Find(∞,R1) Delete(T) MergeWithRoot(R1,R2,T ) return T

Time O(h). 

AVLTreeMergeWithRoot(R1,R2,T )

if |R1.Height− R2.Height| ≤1:   MergeWithRoot(R1,R2,T )   T .Ht← max(R1.Height,R2.Height) +1   return T

Split:

Split(R, x)

if R =null:   return (null,null) if x ≤R.Key:   (R1,R2)← Split(R.Left,x)   R3← MergeWithRoot(R2,R.Right,R)   return (R1,R3) if x >R.Key:   ...

Summary

Merge combines trees. Split turns one tree into two. Both can be implemented inO(log(n))time for AVL trees.