AVL Tree

1
AVL Tree

• ????????????? Binary Tree ???????????? (Balance)
  ????????????????? Subtree ??????????????????
• ?????? Tree ??????????????? Root ??? Tree
  ?????????????????????

2
AVL Tree (Adelson-Velskii and Landis)

• ???? Binary Search Tree ??????????????????????????
  ??????? (Balance Condition)
• Depth ??? Tree ???? O(Log N)
• ??? Height ??? Subtree ???????????????????????????
  ???????? 1

h 2
h 0
3
AVL Tree Operations

• ?????????????? AVL Tree ?????????????? O(Log N)
• ??? Insert ??????????????????????? Update ?????
  Balance ??? Node ??? Node ????????? path
  ????????? Root
• ??? Update ??? Balance ??? Tree
  ???????????????Rotation (???????)
• Single Rotation
• Double Rotation

4
AVL Tree Rotation

• ni ???? Node ?????????????????????????????
  Rebalance ???????????????????????????? Tree
  ???????????? 4 ????
• LL Insert into left subtree of the left child

4
2
1
5
AVL Tree Rotation

• RR Insert into right subtree of the right child

4
6
7
6
AVL Tree Rotation

• LR Insert into right subtree of the left
  subtree of the node

8
4
6
5
7
AVL Tree Rotation

• RL Insert into left subtree of the right
  subtree of the node

8
12
10
11
8
Single Rotation

• ?????? LL ??? RR ??? Insert ??????????????
  (Outside) ??? Tree (????-????, ???-???)
• Rebalance ??????? Single Rotation

LL Insert ??? X ????? k2 ?????????
9
Single Rotation

• RR - Insert ??? Z ????? k1 ?????????

10

• Node 23 ????? subtree ??????????? 16 ?? height 2
  ?????? subtree ?????????? 16 ?? height 0
• Rebalance Node 16 ???? Single Rotation

11

• ????? insert ??? 5, 6 ??? 7 ???? ????? subtree
  ??????????? 5 ?? height2 ?????? subtree
  ????????? height0
• Rebalance Node 5 ???? Single Rotation

12

• insert key 8, 9 ???? ????? subtree ?????????? 7
  ?? height2 ?????? subtree ?????????? height0
• Rebalance Node 7 ???? Single Rotation

13

• ????? insert key 4, 3 ???? ????? subtree
  ?????????? 5 ?? height2 ?????? subtree
  ?????????? height 0
• Rebalance Node 5 ???? Single Rotation

14
Double Rotation

• ?????? LR ??? RL ??? Insert ????????????????
  (Inside) ??? Tree (????-???, ???-????)
• Rebalance ??????? Double Rotation

LR Insert ??? X ????? k2 ????????? Single
Rotation ???????? k1 ?????????
15
Double Rotation

• Left-Right Double Rotaion ???????? LR

16
Double Rotation

• Right-Left Double Rotaion ???????? RL

17

• insert Key 1 ??? 2 ?????? ????????? Rebalance 3
  (LR)
• Rebalance 7 ???? Double Rotations

18

• Insert Key 14, 15 ?????? ????????? Rebalance 9
  (RR)
• Rebalance 9 ???? Single Rotations

19

• Insert Key 10 ?????? ????????? Rebalance 8 (RL)
• Rebalance 8 ???? Double Rotation

20

• Insert Key 11, 12 ?????? ????????? Rebalance 10
  (RR)
• Rebalance 10 ???? Single Rotations