In AVL trees, balance factors are super important for keeping the trees balanced.
So, what exactly is a balance factor?
It’s the difference between the heights of a node's left and right subtrees. We can think of it this way:
Balance Factor = Height of Left Subtree - Height of Right Subtree
The balance factor can be one of three numbers: -1, 0, or 1.
A balance factor of 0 means that the left and right subtrees are the same height. This means the tree is perfectly balanced at that point.
A balance factor of -1 means that the right subtree is one level taller than the left subtree.
A balance factor of 1 shows that the left subtree is one level taller than the right subtree.
Keeping these balance factors right is important when we add or remove nodes from the AVL tree.
If a node's balance factor goes outside the range of -1 to 1 because of an operation, we need rotations to fix the balance. There are four types of rotations based on the kind of imbalance:
After we add or remove a node, we need to check the balance factors of all the nodes above it. If any node has a balance factor of -2 or 2, we have to do rotations to fix it.
The cool thing about AVL trees is that they keep their height very small, around log(n) where n is the number of nodes. This means that searching, adding, or removing a node takes a nice average time of O(log n). On the other hand, unbalanced trees can take much longer, going up to O(n) at their worst.
To sum it up, balance factors are super important for AVL trees. They help us keep the tree balanced using rotations. This clever design ensures that AVL trees are a great choice when we need a data structure that works quickly and efficiently.
In AVL trees, balance factors are super important for keeping the trees balanced.
So, what exactly is a balance factor?
It’s the difference between the heights of a node's left and right subtrees. We can think of it this way:
Balance Factor = Height of Left Subtree - Height of Right Subtree
The balance factor can be one of three numbers: -1, 0, or 1.
A balance factor of 0 means that the left and right subtrees are the same height. This means the tree is perfectly balanced at that point.
A balance factor of -1 means that the right subtree is one level taller than the left subtree.
A balance factor of 1 shows that the left subtree is one level taller than the right subtree.
Keeping these balance factors right is important when we add or remove nodes from the AVL tree.
If a node's balance factor goes outside the range of -1 to 1 because of an operation, we need rotations to fix the balance. There are four types of rotations based on the kind of imbalance:
After we add or remove a node, we need to check the balance factors of all the nodes above it. If any node has a balance factor of -2 or 2, we have to do rotations to fix it.
The cool thing about AVL trees is that they keep their height very small, around log(n) where n is the number of nodes. This means that searching, adding, or removing a node takes a nice average time of O(log n). On the other hand, unbalanced trees can take much longer, going up to O(n) at their worst.
To sum it up, balance factors are super important for AVL trees. They help us keep the tree balanced using rotations. This clever design ensures that AVL trees are a great choice when we need a data structure that works quickly and efficiently.