• A tree is weight balanced if: (size(l) - height(r)| < 2
  
• A tree is height balanced if: |height(l) - height(r)| < 2
  

Tree Rotation

Operations that change the balance of a tree whilst maintaining a search tree

Left Rotate

left subtree moves down
right subtree promoted
left child of right subtree becomes right child of left subtree

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btree_node *btree_rotate_left(btree_node *n1) {
  if (!n1) return NULL;

  btree_node *n2 = n1->right;
  if (!n2) return n1;

  n1->right = n2->left;
  n2->left = n1;

  return n2;
}

Right Rotate

right subtree moves down
left subtree promoted
right child of left subtree becomes left child of right subtree

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2

  B    ->    A
A   C  ->  C   B

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btree_node *btree_rotate_right(btree_node *n1) {
  if (!n1) return NULL;

  btree_node *n2 = n1->left;
  if (!n2) return n1;

  n1->left = n2->right;
  n2->right = n1;

  return n2;
}

Rebalance cost
Many O(n)
Degenerate O(n logn)

When to rebalance
Rebalance on every inseration
or rebalance every k values
or rebalance when a threshold is passed

// Amortisation - more work now less work later