1. 3.1 Height-Balanced Trees 3.2 Weight-Balanced Trees
Chen Song

2. Height-Balanced Trees

3. Height-Balanced Trees
Height – The maximum length of any path from the root to a leaf. Height-Balanced Tree – In each interior node, the height of the right subtree and left subtree differ by at most one.

4. Height-Balanced Trees
Example:

5. Height-Balanced Trees
Theorem
A height-balanced tree of height h has at least
( ) ( ) ℎ − ( 3− ) ( 1− ) ℎ
leaves.

6. Height-Balanced Trees
Tree T
Tree T has at least leaves: leaves(h)=leaves(h-1)+leaves(h-2) leaves(0)=1 leaves(1)=2 Characteristic equation: xh=xh-1+xh-2 => x2-x-1=0

7. Height-Balanced Trees
Insert and Delete
After insert or delete:
Case 1: Tree is still balanced
Case 2: Tree is not balanced at node n
|n->left-height − n->right-height|=2

8. Height-Balanced Trees
1. n->left->height = n->right->height+2 and n->left->left->height=n->right->height+1 Right rotation on node n 2. n->left->height = n->right->height+2 and n->left->left->height=n->right-height Left rotation on node n->left, and follow right rotation on node n

9. Height-Balanced Trees
3. n-> right->height = n->left->height+2 and n->right->right->height=n->left->height+1 Left rotation on node n->left 4. n->right->height = n->left->height+2 and n->right->right->height=n->left-height Right rotation on node n->right, and follow left rotation on node n

10. Height-Balanced Trees
Height-Balanced Tree structure supports search, insert, and delete in O(logn) time
Search => O(logn)
Insert => search + insert + rebalance => O(logn)
O(logn) O(1) O(logn)
Delete => search + delete + rebalance => O(logn)
O(logn) O(1) O(logn)

11. Weight-Balanced Trees

12. Weight-Balanced Trees
Weight – The number of leaves of a tree. Weight-Balanced Tree – The weight of the right and left subtree in each node differ by at most one.

13. Weight-Balanced Trees
For each subtree, the left and right sub-subtrees has each at least a fraction of α of total weight of the subtree.
αWT≤WT1≤(1-α)WT
αWT≤WT2≤(1-α)WT
Tree T: T1 T2

14. Weight-Balanced Trees
Theorem
An α-weight-balanced tree of height h≥2 has at least
( 1 1−α ) ℎ leaves.

15. Weight-Balanced Trees
Rebalance
n is current node α∈[ 2 7 , 1− ]
Case 1:
n->left->weight ≥ α*n->weight and
n->right->weight ≥ α*n->weight
No rebalancing

16. Weight-Balanced Trees
Case 2: n->right->weight ≥ α*n->weight If n->left->left->weight > (α+ε)n->weight, do right rotation on node n Else left rotation on n->left, followed right rotation on node n. ε ≤ α2-2α+ 1 2

17. Weight-Balanced Trees
Case 3: n->left->weight ≥ α*n->weight If n->right->right->weight > (α+ε)n->weight, do left rotation on node n Else right rotation on n->right, followed left rotation on node n.

18. Weight-Balanced Trees
Theorem
The weight-balanced tree structure supports search, insert, and delete in O(logn) time.
Search => O(logn)
Insert => search + insert + rebalance => O(logn)
O(logn) O(1) O(logn)
Delete => search + delete + rebalance => O(logn)
O(logn) O(1) O(logn)

19. END