Properties of BST delete - PowerPoint Presentation

Slide1

Properties of BST delete

We first do the normal BST deletion:0 children: just delete it1 child: delete it, connect child to parent2 children: put successor in your place, delete successorWhich nodes’ heights may have changed:0 children: path from deleted node to root1 child: path from deleted node to root2 children: path from deleted successor to rootWill rebalance as we return along the “path in question” to the root

1

CSE332: Data Abstractions

20

9

2

15

5

12

7

10

Case #1 Left-left due to right deletion

CSE332: Data Abstractions

2

Start with some

subtree where if right child Z becomes shorter we are unbalanced due to height of left-left grandchild X

A delete in the right child Z could cause this right-side shortening

h

a

Z

Y

b

X

h+1

h

h+1

h+2

h+3

Case #1: Left-left due to right deletion

CSE332: Data Abstractions

3

h

a

Z

Y

b

X

h+1

h

h+1

h+2

h+3

b

Z

Y

a

h+1

h

h+1

h+2

X

h

h+1

Same single rotation as when an insert in the left-left grandchild X caused imbalance due to X becoming taller

But here the “height” at the top decreases, so more rebalancing farther up the tree might still be necessary

Case #2: Left-right due to right deletion

a

h-1

h

h

V

U

h+1

h+2

h+3

Z

X

b

c

h+1

h

c

X

h-1

h+1

h

h+1

V

U

h+2

Z

a

b

h

h+1

h

Same double rotation when an insert in the left-right grandchild caused imbalance due to c becoming taller

But here the “height” at the top decreases, so more rebalancing farther up the tree might still be necessary

No third right-deletion case needed

So far we have handled these two cases:left-left left-right

5

CSE332: Data Abstractions

h

a

Z

Y

b

X

h+1

h

h+1

h+2

h+3

a

h-1

h

h

V

U

h+1

h+2

h+3

Z

X

b

c

h+1

h

But what

if the two left grandchildren X & Y are now

both

too tall (h+1)?

Then it turns out

left-left solution still works

The children of the “new top node” will have heights differing by 1 instead of 0, but that’s fine

Mirror Image cases

For each of the cases described above, we can also have the mirror image: deletion in the left side of the tree, and needing to compare the heights of the right grandchildren